What is the purpose of finding an average: What is the average? | TheSchoolRun
Posted onAverage vs Weighted Average | Top 4 Differences (with Infographics)
Average and weighted average are two different terms in Excel. Average is a method to calculate the central point of a given data set. The traditional way of calculating an average of adding the numbers divided by the number of data sets is estimated. In contrast, a weighted average is calculated similarly but with a weight multiplied by each data set.
For example, a company owner purchases 40,000 units of a product at $5 each and 20,000 at $7 each. We will use these units as the weight and the total number of units as the sum of all weights to calculate the weighted average. So, we will get,
=$5(40,000) +$7(20,000)/40,000+20,000
= ($200,000+$140,000)/60,000
= $160,000/60,000
=$5.66
So, the average weighted cost is $5.66 per unit.
With the same example, if we can calculate an average. It can be calculated by adding all of them and dividing the same by several observations.
Table of contents
- Difference Between Average vs Weighted Average
- What is Average?
- Example of Average
- What is the Weighted Average?
- Example of Weighted Average
- Average vs Weighted Average Infographics
- Average vs Weighted Average – Key Differences
- Average vs. Weighted Average Head-to-Head Differences
- Conclusion
- Recommended Articles
The average and weighted average are mathematical and statistical terms in finance and business. But both are calculated differently. The average is the sum of all individual observations divided by the number of observations. Average is used to find the middle value in a particular data set. It is also known as a central tendency. It is used to find the central tendencyCentral TendencyCentral Tendency is a statistical measure that displays the centre point of the entire Data Distribution & you can find it using 3 different measures, i.e., Mean, Median, & Mode. read more of a group of data in a specific group of data. The weighted average is used in the field of accounting. The primary purpose is to find the right weight or value to solve. For example, the weighted average is the value of the principal repayment of certain bonds or loans until the principal value is paid.
What is Average?
The average is the sum of all individual observations divided by the number of observations. It is used to find the middle value in a particular data set. It is also known as a central tendency. It is used to find the central tendency of a group of data in a specific group of data. It is mainly used for the representation of data. We can solve it for a data set by using the arithmetic formula.
Average formula = Sum of Observation / Number of Observation
Example of Average
Let us see an example to understand the average.
Suppose ten students in class score 50, 60, 70, 80, 65, 78, 95, 63, 58, 91, respectively, out of 100. Now, let us find the average for the above marks of a student.
Average formula = Sum of Observation / Number of Observation
Sum of Observation = 50 + 60 + 70 + 80 + 65 + 78 + 95 + 63 + 58 + 91
So, the average of the class of 10 students is 71.
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What is the Weighted Average?
The weighted average is used in the field of accounting. The main purpose is to find the right weight or value to solve. The weighted average is the value of the principal repayment of certain bonds or loans until the principal value is paid. The weighted average is also a type of average with a small difference. All observations do not carry equal weights; different observations carry different importance. Each observation is multiplied by the weight and added up. Weight average is used to perform. It can be taken as an average in which every value has a different weight. And it is influenced by the weight of the data value. The weighted value is the sum of the product of observation into weight divided by the sum of weight and can be written as: –
Weighted Average Formula = (a1w1 + a2w2 + a3w3 + …+ anwn) / (w1 + w2 + w3 … +wn)
Example of Weighted Average
Let us see an example to understand it better.
Suppose three different exams contribute to giving final marks for a year. There is a different weight for each exam. For example, for the first exam, the weight was 15%, for the second exam, the weight was 25%, and for the final exam, the weight was 60%. Now, let us assume a student scored 60 marks in the first, 70 in the second, and 80 in the final exam out of 100. Let us calculate the final marks of a student.
Use the formula mentioned above for its calculation.
- So, the weighted average of a student is 74. 5.
Average vs Weighted Average Infographics
Here, we provide you with the top 5 differences.
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Average vs Weighted Average – Key Differences
The key differences between this average and weighted average are as follows: –
- The average is the sum of all individual observations divided by the number of observations. In contrast, the weighted average is observation multiplied by the weight and added to find a solution.
- An average is a mathematical equation, whereas the weighted average is applied in the daily activities of finance.
- The average represents a set of data. At the same time, the weighted average needs to be evaluated to solve a problem.
- One can solve the average for the data set by using the arithmetic formula. The weighted average component is given the weight of value to arrive at a specific answer.
Average vs. Weighted Average Head-to-Head Differences
Let us now look at the head-to-head differences.
Basis | Average | Weighted Average |
---|---|---|
Definition | It is the sum of all individual observations divided by the number of observations. | Observation is multiplied by the weight and added up to find a solution. |
Equation | It is a mathematical equation. | It is applied in the daily activities of finance. |
Solution | It is a representation of a set of data. | It needs to be evaluated to solve a problem. |
Calculations | We can solve it for a data set using the arithmetic formula. | The component is given the weight of value to arrive at a specific answer. |
Conclusion
So, we have seen the average and weighted average and the difference between the two. We have seen that the average is the sum of all individual observations divided by the number of observations. We can solve the average for a data set by using an arithmetic formula. The weighted average is observation multiplied by a weight and added up to find a solution. The weighted average component is given a weight of value to arrive at a specific answer. Based on the problem, both have different users, and both are computed differently. The main purpose of the weighted average is to find the right weight or value to solve. For example, the weighted average is the average value of the principal repayment of certain bonds or loans until a principal value is paid. An average is used to find the median value or average value.
Recommended Articles
This article is a guide to the Average vs. Weighted Average. Here, we discuss the top differences between Average vs. Weighted Average and infographics and a comparison table. You may also have a look at the following articles: –
- EWMA
- Average Formula in Excel
- Square Root in Excel
- INT Excel Function
What Is It, How Is It Calculated and Used?
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What Is Weighted Average?
Weighted average is a calculation that takes into account the varying degrees of importance of the numbers in a data set. In calculating a weighted average, each number in the data set is multiplied by a predetermined weight before the final calculation is made.
A weighted average can be more accurate than a simple average in which all numbers in a data set are assigned an identical weight.
Key Takeaways
- The weighted average takes into account the relative importance or frequency of some factors in a data set.
- A weighted average is sometimes more accurate than a simple average.
- In a weighted average, each data point value is multiplied by the assigned weight, which is then summed and divided by the number of data points.
- For this reason, a weighted average can improve the data’s accuracy.
- Stock investors use a weighted average to track the cost basis of shares bought at varying times.
Weighted Average
What Is the Purpose of a Weighted Average?
In calculating a simple average, or arithmetic mean, all numbers are treated equally and assigned equal weight. But a weighted average assigns weights that determine in advance the relative importance of each data point.
A weighted average is most often computed to equalize the frequency of the values in a data set. For example, a survey may gather enough responses from every age group to be considered statistically valid, but the 18–34 age group may have fewer respondents than all others relative to their share of the population. The survey team may weight the results of the 18–34 age group so that their views are represented proportionately.
However, values in a data set may be weighted for other reasons than the frequency of occurrence. For example, if students in a dance class are graded on skill, attendance, and manners, the grade for skill may be given greater weight than the other factors.
In any case, in a weighted average, each data point value is multiplied by the assigned weight, which is then summed and divided by the number of data points.
In a weighted average, the final average number reflects the relative importance of each observation and is thus more descriptive than a simple average. It also has the effect of smoothing out the data and enhancing its accuracy.
Weighted Average | |||
---|---|---|---|
Data Point | Data Point Value | Assigned Weight | Data Point Weighted Value |
1 | 10 | 2 | 20 |
1 | 50 | 5 | 250 |
1 | 40 | 3 | 120 |
TOTAL | 100 | 10 | 390 |
Weighted Average | 39 |
Weighting a Stock Portfolio
Investors usually build a position in a stock over a period of several years. That makes it tough to keep track of the cost basis on those shares and their relative changes in value.
The investor can calculate a weighted average of the share price paid for the shares. To do so, multiply the number of shares acquired at each price by that price, add those values, then divide the total value by the total number of shares.
A weighted average is arrived at by determining in advance the relative importance of each data point.
For example, say an investor acquires 100 shares of a company in year one at $10, and 50 shares of the same stock in year two at $40. To get a weighted average of the price paid, the investor multiplies 100 shares by $10 for year one and 50 shares by $40 for year two, then adds the results to get a total of $3,000. Then the total amount paid for the shares, $3,000 in this case, is divided by the number of shares acquired over both years, 150, to get the weighted average price paid of $20.
This average is now weighted with respect to the number of shares acquired at each price, not just the absolute price.
Examples of Weighted Averages
Weighted averages show up in many areas of finance besides the purchase price of shares, including portfolio returns, inventory accounting, and valuation.
When a fund that holds multiple securities is up 10 percent on the year, that 10 percent represents a weighted average of returns for the fund with respect to the value of each position in the fund.
For inventory accounting, the weighted average value of inventory accounts for fluctuations in commodity prices, for example, while LIFO (last in, first out) or FIFO (first in, first out) methods give more importance to time than value.
When evaluating companies to discern whether their shares are correctly priced, investors use the weighted average cost of capital (WACC) to discount a company’s cash flows. WACC is weighted based on the market value of debt and equity in a company’s capital structure.
How does a weighted average differ from a simple average?
A weighted average accounts for the relative contribution, or weight, of the things being averaged, while a simple average does not. Therefore, it gives more value to those items in the average that occur relatively more.
What are some examples of weighted averages used In finance?
Many weighted averages are found in finance, including the volume-weighted average price (VWAP), the weighted average cost of capital (WACC), and exponential moving averages (EMAs) used in charting. Construction of portfolio weights and the LIFO (last in, first out) and FIFO (first in, first out) inventory methods also make use of weighted averages.
How is a weighted average calculated?
You can compute a weighted average by multiplying its relative proportion or percentage by its value in sequence and adding those sums together. Thus, if a portfolio is made up of 55% stocks, 40% bonds, and 5% cash, those weights would be multiplied by their annual performance to get a weighted average return. So if stocks, bonds, and cash returned 10%, 5%, and 2%, respectively, the weighted average return would be (55 × 10%) + (40 × 5%) + (5 × 2%) = 7. 6%.
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-
Tax Foundation. “Inventory Valuation in Europe.”
-
My Accounting Course. “Weighted Average Cost of Capital (WACC) Guide.”
20011 — Measures of Central Tendency: Mean, Median, and Mode
Introduction: Connecting Your Learning
In real-world applications, you can use tables and graphs of various kinds to show information and to extract information from data that can lead to analyses and predictions. Graphs allow you to communicate a message from data.
Measures of central tendency are a key way to discuss and communicate with graphs. The term central tendency refers to the middle, or typical, value of a set of data, which is most commonly measured by using the three m’s: mean, median, and mode. The mean, median, and mode are known as the measures of central tendency. In this lesson, you will explore these three concepts.
Focusing Your Learning
Lesson Objectives
By the end of this lesson, you should be able to:
- Compute the mean, median, and mode of a given set of data.
- Identify an outlier given a set of data.
- Identify the mode or modes of a data set for both quantitative and qualitative data.
Key Terms
Presentation
The Mean, Median, and Mode
Mean, median, and mode are three basic ways to look at the value of a set of numbers. You will start by learning about the mean.
The mean, often called the average, of a numerical set of data, is simply the sum of the data values divided by the number of values. This is also referred to as the arithmetic mean. The mean is the balance point of a distribution.
Mean = |
sum of the values |
the number of values |
For instance, take a look at the following example. Use the formula to calculate the mean number of hours that Stephen worked each month based on the example below.
Example | |
Problem |
Stephen has been working on programing and updating a Web site for his company for the past 15 months. The following numbers represent the number of hours Stephen has worked on this Web site for each of the past 7 months:
24, 25, 31, 50, 53, 66, 78
What is the mean (average) number of hours that Stephen worked on this Web site each month? |
Step 1: Add the numbers to determine the total number of hours he worked.
24 + 25 + 33 + 50 + 53 + 66 + 78 = 329 |
|
Step 2: Divide the total by the number of months.
|
|
Answer
The mean number of hours that Stephen worked each month was 47. |
The calculations for the mean of a sample and the total population are done in the same way. However, the mean of a population is constant, while the mean of a sample varies from sample to sample.
Example | |
Problem |
Mark operates Technology Titans, a Web site service that employs 8 people. Find the mean age of his workers if the ages of the employees are as follows:
55, 63, 34, 59, 29, 46, 51, 41 |
Step 1: Add the numbers to determine the total age of the workers.
Step 2: Divide the total by the number of months.
|
|
Answer
The mean age of all 8 employees is 47.25 years, or 47 years and 3 months. |
Look at another approach. If you were to take a sample of 3 employees from the group of 8 and calculate the mean age for these 3 workers, would the results change?
Use the ages 55, 29, and 46 for one sample of 3, and the ages 34, 41, and 59 for another sample of 3:
The mean age of the first group of 3 employees is 43.33 years.
The mean age of the second group of 3 employees is 44.66 years.
The mean age for a sample of a population depends upon the values that are included in the sample. From this example, you can see that the mean of a population and that of a sample from the population are not necessarily the same.
In addition to calculating the mean for a given set of data values, you can apply your understanding of the mean to determine other information that may be asked for in everyday problems.
Example | |
Problem |
Two weeks before Mark opened Technology Titans, he launched his company Web site. During those 14 days, Mark had an average of 24.5 hits on his Web site per day. In the first two days that Technology Titans was open for business, the Web site received 42 and 53 hits respectively. Determine the new average for hits on the Web site. |
Step 1: Multiply the given average by 14 to determine the total number of hits on Mark’s Web site.
Step 2: Add the hits for the first two days his business was open.
Step 3: Divide this new total by 16 to determine the new average.
|
|
Answer
The average number of hits Mark’s Web site has received per day since it was launched is 27.375. |
All values for the means you have calculated so far have been for ungrouped, or listed, data. A mean can also be determined for data that is grouped, or placed in intervals. Unlike listed data, the individual values for grouped data are not available, and you are not able to calculate their sum. To calculate the mean of grouped data, the first step is to determine the midpoint of each interval or class. These midpoints must then be multiplied by the frequencies of the corresponding classes. The sum of the products divided by the total number of values will be the value of the mean.
The following example will show how the mean value for grouped data can be calculated.
Example | |||
Problem |
In Tim’s office, there are 25 employees. Each employee travels to work every morning in his or her own car. The distribution of the driving times (in minutes) from home to work for the employees is shown in the table below. |
||
|
|||
Calculate the mean of the driving times. |
|||
Step 1: Determine the midpoint for each interval.
For 0 to less than 10, the midpoint is 5.
For 10 to less than 20, the midpoint is 15.
For 20 to less than 30, the midpoint is 25.
For 30 to less than 40, the midpoint is 35.
For 40 to less than 50, the midpoint is 45.
Step 2: Multiply each midpoint by the frequency for the class.
For 0 to less than 10, (5)(3) = 15
For 10 to less than 20, (15)(10) = 150
For 20 to less than 30, (25)(6) = 150
For 30 to less than 40, (35)(4) = 140
For 40 to less than 50, (45)(2) = 90
Step 3: Add the results from Step 2 and divide the sum by 25.
15 + 150 + 150 + 140 + 90 = 545
|
|||
Answer
Each employee spends an average (mean) time of 21.8 minutes driving from home to work each morning. |
The mean is often used as a summary statistic. However, it is affected by extreme values (outliers): either an unusually high or low number. When you have extreme values at one end of a data set, the mean is not a very good summary statistic.
Example: Outliers
If you were employed by a company that paid all of its employees a salary between $60,000 and $70,000, you could probably estimate the mean salary to be about $65,000.
However, if you had to add in the $150,000 salary of the CEO when calculating the mean, then the value of the mean would increase greatly. It would, in fact, be the mean of the employees’ salaries, but it probably would not be a good measure of the central tendency of the salaries.
In addition to calculating the mean for a given set of data values, you can also apply your understanding of the mean to determine other information that may be asked for in everyday problems.
The Median
What is the Median?
The median is the number that falls in the middle position once the data has been organized. Organized data means the numbers are arranged from smallest to largest or from largest to smallest. The median for an odd number of data values is the value that divides the data into two halves. If n represents the number of data values and n is an odd number, then the median will be found in the position.
This measure of central tendency is typically used when the mean value is affected by an unusually low number or an unusually high number in the data set (outliers). Outliers distort the mean value to the extent that the mean value no longer accurately depicts the set of data.
For example: If one of the houses in your neighborhood was broken down and maintained a low property value, then you would not want to include this property when determining the value of your own home. However, if you are purchasing a home in that neighborhood, you may want to include the outlier since it would drive down the price you would have to pay.
Try a few examples to follow the steps needed to calculate the median.
Example | |
Problem |
Find the median of the following data:
|
Step 1: Organize the data, or arrange the numbers from smallest to largest.
Step 2: Since the number of data values is odd, the median will be found in the position.
Step 3: In this case, the median is the value that is found in the fourth position of the organized data.
|
|
Answer
The median is 10. |
Another way to look at the example is to narrow the data down to find the middle number.
2, 6, 8, 10, 12, 14, 16
Χ, 6, 8, 10, 12, 14, Χ
Χ, Χ, 8, 10, 12, Χ, Χ
Χ, Χ, Χ, 10, Χ, Χ, Χ
Here is another example of how to calculate the median of a set of numbers.
Example | |
Problem |
Find the median of the following data:
|
Step 1: Organize the data, or arrange the numbers from smallest to largest.
Step 2: Since the number of data values is even, the median will be the mean value of the numbers found before and after the position.
Step 3: The number found before the 5.5 position is 4 and the number found after the 5.5 position is 6. Now, you need to find the mean value.
|
|
Answer
The median is 5. |
The Mode
What is the Mode?
The mode of a set of data is simply the value that appears most frequently in the set.
If two or more values appear with the same frequency, each is a mode. The downside to using the mode as a measure of central tendency is that a set of data may have no mode, or it may have more than one mode. However, the same set of data will have only one mean and only one median.
- The word modal is often used when referring to the mode of a data set.
- If a data set has only one value that occurs most often, the set is called unimodal.
- A data set that has two values that occur with the same greatest frequency is referred to as bimodal.
- When a set of data has more than two values that occur with the same greatest frequency, the set is called multimodal.
When determining the mode of a data set, calculations are not required, but keen observation is a must. The mode is a measure of central tendency that is simple to locate, but it is not used much in practical applications.
Example | |
Problem |
Find the mode of the following data:
76, 81, 79, 80, 78, 83, 77, 79, 82, 75 |
There is no need to organize the data, unless you think that it would be easier to locate the mode if the numbers were arranged from least to greatest. In the above data set, the number 79 appears twice, but all the other numbers appear only once. Since 79 appears with the greatest frequency, it is the mode of the data values. |
|
Answer
The mode is 79. |
Example | |
Problem |
The ages of 12 randomly selected customers at a local Best Buy are listed below:
23, 21, 29, 24, 31, 21, 27, 23, 24, 32, 33, 19
What is the mode of the above ages? |
The above data set has three values that each occur with a frequency of 2. These values are 21, 23, and 24. All other values occur only once. Therefore, this set of data has three modes. |
|
Answer
The modes are 21, 23, and 24. |
Remember that the mode can be determined for qualitative data as well as quantitative data, but the mean and the median can only be determined for quantitative data.
Now that you have added to your knowledge by reviewing the lesson and the examples, it is time to watch the following Khan Academy videos. These videos will provide additional explanations and working examples of how to determine the mean, median, and mode to help you gain a better understanding of this new concept.
In this lesson, you have learned how to calculate the mean, median, and mode of a set of data values. In addition, you have been introduced to other key terms such as measures of central tendency, unimodal, bimodal, and outliers. You also learned that the mode is the only measure of central tendency used in both quantitative and qualitative data.
As with every lesson and module, you are encouraged to research how these topics pertain to your particular area of study within the world of information technology. By now you are very aware that not every topic in mathematics will be directly implemented in your future career field. However, do not rule out the possibility that this topic might be an integral part of your future until you do some research.
“Chapter 5: Measures of Central Tendency” by Merry, B. © 2012 retrieved from http://www.ck12.org/flexbook/chapter/9079 and used under a Creative Commons Attribution http://creativecommons.org/licenses/by/3.0/. This is an adaption of the lesson titled, “Measures of Central Tendency: Mean, Median, and Mode” by the National Information Security and Geospatial Technologies Consortium (NISGTC) is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0
How did the concept of the mean come about?
In 1906, the great scientist and renowned eugenicist Francis Galton visited the annual Animal and Poultry Exhibition in western England, where, quite by chance, he performed an interesting experiment.
According to James Surowetsky, author of The Wisdom of the Crowd, there was a competition at the Galton Fair in which people had to guess the weight of a slaughtered bull. The one who named the closest to the true number was declared the winner.
Galton was known for his contempt for the intellectual abilities of ordinary people. He believed that only real experts would be able to make accurate statements about the bull’s weight. And 787 participants of the competition were not experts.
The scientist was going to prove the incompetence of the crowd by calculating the average from the participants’ answers. What was his surprise when it turned out that the result he received corresponded almost exactly to the real weight of the bull!
- The brewer who revolutionized applied statistics
Average value — late invention
Of course, the accuracy of the answer amazed the researcher. But even more remarkable is the fact that Galton thought of using the average at all.
In today’s world, averages and so-called medians are everywhere: the average temperature in New York in April is 52 degrees Fahrenheit; Stephen Curry averages 30 points per game; median household income in the US is $51 939/year.
However, the idea that many different outcomes can be represented by a single number is quite new. Until the 17th century, averages were not generally used.
How did the concept of mean and median values appear and develop? And how did it manage to become the main measuring technique in our time?
The predominance of means over medians had far-reaching consequences for our understanding of information. And often it led people astray.
Mean and median
Imagine you are telling a story about four people who dined with you last night at a restaurant. You would give one of them 20 years, another 30, the third 40, and the fourth 50. What would you say about their ages in your story?
Most likely, you will name their average age.
The mean is often used to convey information about something, as well as to describe a set of measurements. Technically, the average is what mathematicians call the «arithmetic mean» — the sum of all measurements divided by the number of measurements.
Although the word «average» is often used as a synonym for the word «median» (median), the latter is more often used to refer to the middle of something. This word comes from the Latin «medianus», which means «middle».
- Interesting statistics
Median value in ancient Greece
The history of the median value originates from the teachings of the ancient Greek mathematician Pythagoras. For Pythagoras and his school, the median had a clear definition and was very different from how we understand the average today. It was used only in mathematics, not in data analysis.
In the Pythagorean school, the median value was the average number in a three-term sequence of numbers, in «equal» relation to neighboring terms. «Equal» ratio could mean the same distance. For example, the number 4 in the row 2,4,6. However, it could also express a geometric progression, such as 10 in the sequence 1,10,100.
The statistician Churchill Eisenhart explains that in ancient Greece, the median was not used to represent or stand in for any set of numbers. It simply denoted the middle, and was often used in mathematical proofs.
Eisenhart spent ten years studying the mean and median. Initially, he tried to find the representative function of the median in early scientific constructions. Instead, however, he found that most of the early physicists and astronomers relied on single, skillfully made measurements, and they did not have a methodology to choose the best result among many observations.
Modern researchers base their conclusions on the collection of large amounts of data, such as biologists studying the human genome. Ancient scientists, on the other hand, could take several measurements, but chose only the best for building their theories.
As the historian of astronomy Otto Neugebauer wrote, «this is consistent with the conscious desire of ancient people to minimize the amount of empirical data in science, because they did not believe in the accuracy of direct observations.»
For example, the Greek mathematician and astronomer Ptolemy calculated the angular diameter of the Moon using the method of observation and the theory of the earth’s motion. His score was 31’20. Today we know that the diameter of the Moon ranges from 29’20 to 34’6, depending on the distance from the Earth. Ptolemy used little data in his calculations, but he had every reason to believe that they were accurate.
Eisenhart writes: “It must be borne in mind that the connection between observation and theory in antiquity was different than it is today. The results of observations were understood not as facts to which a theory must conform, but as concrete cases that can only be useful as illustrative examples of the truth of a theory. values were not used in this role. From antiquity to the present day, another mathematical concept has been used as such a representative means — the half-sum of extreme values.
- Interesting statistics. Part two, gigantic
Half sum of extremes
New scientific tools almost always arise from the need to solve a certain problem in some discipline. The need to find the best value among many measurements arose from the need to accurately determine the geographic location.
The 11th century intellectual giant Al-Biruni is known as one of the first people to use the methodology of representative meanings. Al-Biruni wrote that when he had many measurements at his disposal and wanted to find the best among them, he used the following «rule»: you need to find a number corresponding to the middle between two extreme values. When calculating the half-sum of extreme values, all numbers between the maximum and minimum values \u200b\u200bare not taken into account, but only the average of these two numbers is found.
Al-Biruni used this method in various fields, including to calculate the longitude of the city of Ghazni, which is located on the territory of modern Afghanistan, as well as in his studies of the properties of metals.
However, in the last few centuries, the half-sum of extreme values has been used less and less. In fact, in modern science, it is not relevant at all. The median value replaced the half-sum.
Moving to Averages
By the early 19th century, the use of the median/mean had become a common method for finding the most accurately representative value from a group of data. Friedrich von Gauss, the outstanding mathematician of his time, in 1809th year wrote: “It was believed that if a certain number was determined by several direct observations made under the same conditions, then the arithmetic mean value is the most true value. If it is not quite strict, then at least it is close to reality, and therefore one can always rely on it.
“Although not exactly strict, the mean is close to reality and can be relied upon. ” – Gauss
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Why did this shift in methodology occur?
This question is rather difficult to answer. In his research, Churchill Eisenhart suggests that the method of finding the arithmetic mean could have originated in the field of measuring magnetic deviation, that is, in finding the difference between the direction of the compass needle pointing north and the real north. This measurement was extremely important during the Age of Discovery.
Eisenhart found that until the end of the 16th century, most scientists who measured magnetic deviation used the ad hoc method (from Latin «to this, for this occasion, for this purpose») in choosing the most accurate measurement.
But in 1580, the scientist William Borough approached the problem differently. He took eight different measurements of deflection and compared them, and concluded that the most accurate reading was between 11 ⅓ and 11 ¼ degrees. He probably calculated the arithmetic mean, which was in this range. However, Borough himself did not openly call his approach the new method.
Until 1635, there were no unequivocal cases of using the average value as a representative number. However, it was then that the English astronomer Henry Gellibrand took two different measurements of the magnetic deflection. One was done in the morning (11 degrees) and the other in the afternoon (11 degrees and 32 minutes). Calculating the most true value, he wrote:
«If we find the arithmetic mean, we can say with high probability that the result of an exact measurement should be about 11 degrees 16 minutes.»
It is likely that this was the first time that the average value was used as the closest to the true!
The word «average» was used in English in the early 16th century to denote financial loss from damage that a ship or cargo suffered during a voyage. For the next hundred years, it denoted precisely these losses, which were calculated as the arithmetic mean. For example, if a ship was damaged during a voyage and the crew had to throw some goods overboard to save the ship’s weight, the investors suffered a financial loss equivalent to the amount of their investment — these losses were calculated in the same way as the arithmetic average. So gradually the values of the average (average) and the arithmetic mean converged.
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Median
Today, the average or arithmetic mean is used as the primary way to select a representative value for multiple measurements. How did it happen? Why was this role not assigned to the median value?
Francis Galton was the champion of the median
The term «median» — the middle term in a series of numbers that divides this series by half — appeared at about the same time as the arithmetic mean. At 159In the 9th year, mathematician Edward Wright, who was working on the problem of normal deviation in a compass, first proposed using the median value.
“…Let’s say a lot of archers are shooting at some target. The target is subsequently removed. How can you find out where the target was? You need to find the middle place between all the arrows. Likewise, among the set of results of observations, the closest to the truth will be the one in the middle.
The median was widely used in the nineteenth century, becoming an indispensable part of any data analysis at that time. It was also used by Francis Galton, the eminent nineteenth-century analyst. In the bull weighing story at the beginning of this article, Galton originally used the median as representing the opinion of the crowd.
Many analysts, including Galton, preferred the median because it is easier to calculate for smaller datasets.
However, the median has never been more popular than the mean. Most likely, this happened due to the special statistical properties inherent in the mean value, as well as its relationship to the normal distribution.
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Relationship between mean and normal distribution
When we take many measurements, the results are, as statisticians say, «normally distributed.» This means that if this data is plotted on a graph, then the points on it will depict something similar to a bell. If you connect them, you get a «bell-shaped» curve. Many statistics fit the normal distribution, such as height of people, IQ, and the highest annual temperature.
When the data is normally distributed, the mean will be very close to the highest point on the bell curve, and a very large number of measurements will be close to the mean. There is even a formula that predicts how many measurements will be some distance from the average.
Thus, the calculation of the average value gives researchers a lot of additional information.
The relationship of the mean to the standard deviation gives it a great advantage, because the median has no such relationship. This connection is an important part of the analysis of experimental data and statistical processing of information. That is why the average has become the core of statistics and all sciences that rely on multiple data for their conclusions.
The advantage of the average is also due to the fact that it is easily calculated by computers. Although the median value for a small group of data is fairly easy to calculate on your own, it is much easier to write a computer program that would find the average value. If you use Microsoft Excel, you probably know that the median function is not as easy to calculate as the mean value function.
As a result, due to its great scientific value and ease of use, the average value became the main representative value. However, this option is not always the best.
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Benefits of the median
In many cases when we want to calculate the center of a distribution, the median is the best measure. This is because the average value is largely determined by the extreme measurements.
Many analysts believe that the thoughtless use of the mean negatively affects our understanding of quantitative information. People look at the average and think it’s «normal». But in fact it can be defined by some one term that stands out strongly from the homogeneous series.
Imagine an analyst who wants to know a representative value for the value of five houses. Four houses are worth $100,000 and the fifth is $900,000. The mean would then be $200,000 and the median would be $100,000. In this, as in many other cases, the median value gives a better understanding of what can be called a «standard».
Recognizing how extremes can affect the average, the median is used to reflect changes in US household income.
The median is also less sensitive to the «dirty» data that analysts deal with today. Many statisticians and analysts collect information by interviewing people on the Internet. If the user accidentally adds an extra zero to the answer, which turns 100 into 1000, then this error will affect the mean much more than the median.
Mean or median?
The choice between the median and the mean has far-reaching implications, from our understanding of the effects of medicines on health to our knowledge of what a family’s standard budget is.
As the collection and analysis of data increasingly determines how we understand the world, so does the importance of the quantities we use. In an ideal world, analysts would use both the mean and median to plot the data.
But we live with limited time and attention. Because of these limitations, we often need to choose just one. And in many cases, the median value is preferable.
High conversions! How to calculate the average value in Excel: formula, functions, tools like calculating the average. Mathematically, this action is performed by dividing the sum of all numbers by their number. How to do it in Excel? Let’s figure it out. 9Ol000 conditional average
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See also: “How to find the inverse of a matrix in Excel”
Perhaps this is the easiest and fastest way to determine the average value. To do this, it is enough to select a range containing two or more cells, and the average value for them will immediately be displayed in the program status bar.
If this information is not available, the corresponding item is most likely disabled in the settings. To turn it back on, right-click on the status bar, in the list that opens, check the checkbox next to line “Medium” . You can set it if necessary with a simple click of the left mouse button.
Calculation of the average value
When the average value needs not only to be determined, but also to be recorded in a separate cell selected for this purpose, several methods can be used. Below we will look at each of them in detail.
Using an arithmetic expression
As we know, the average is equal to the sum of the numbers divided by their number. This formula can also be used in Excel.
- We get up in the desired cell, put the sign “equal” and write an arithmetic expression according to the following principle:
= (Number1 + Number2 + Number3 ...) / Number_of_terms
.
Note: can be either a specific numeric value or a cell reference. In our case, let’s try to calculate the average of the numbers in cells B2, C2, D2 and E2.
The final form of the formula is as follows:=(B2+E2+D2+E2)/4
. - When everything is ready, press Enter to get the result.
This method is certainly good, but the convenience of its use is significantly limited by the amount of data being processed, because it will take a lot of time to list all the numbers or cell coordinates in a large array, moreover, in this case, the possibility of making an error is not ruled out.
Ribbon tools
This method is based on using a special tool on the program ribbon. Here’s how it works:
- We select a range of cells with numerical data for which we want to determine the average value.
- Go to the tab “Main” (if we are not in it). In the tool section “Editing” find the icon “AutoSum” and click on the small down arrow next to it. In the list that opens, click on the option “Average” .
- Immediately below the selected range, the result will be displayed, which is the average value for all the marked cells.
- If we go to the cell with the result, then in the formula bar we will see which function was used by the program for calculations — this is the operator AVERAGE , whose arguments are the range of cells we selected.
Note: If a horizontal selection is made instead of a vertical selection (whole column or part of it), the result will not be displayed under the selection area, but to the right of it.
This method is quite simple and allows you to quickly get the desired result. However, in addition to the obvious pluses, it also has a minus. The fact is that it allows you to calculate the average value only for cells located in a row, moreover, only in one column or row.
To make it clearer, let’s analyze the following situation. Let’s say we have two rows filled with data. We want to get the average value for two rows at once, therefore, we select them and apply the considered tool.
As a result, we will get the average values under each column, which is also not bad if that was the goal.
But if, nevertheless, it is required to determine the average value over several rows / columns or cells scattered in different places in the table, the methods described below will come in handy.
Alternative way to use “Average” on the ribbon:
- Go to the first free cell after the column or row (depending on the data structure) and press the button for calculating the average value.
- Instead of instantly displaying the result, this time the program will offer us to first check the range of cells over which the average value will be calculated, and, if necessary, correct its coordinates.
- Press key 9 when ready0125 Enter and get the result in the given cell.
Using the AVERAGE function
We already got acquainted with this function when we moved to the cell with the result of calculating the average value. Now let’s learn how to fully use it.
- We get into the cell where we plan to display the result. Click on the icon “Insert Function” (fx) to the left of the formula bar.
- In the window that opens Function Wizard select category “Statistical” , in the proposed list, click on the line “AVERAGE” , then click OK .
- A window with function arguments will be displayed on the screen (their maximum number is 255). Specify the coordinates of the desired range as the value of the argument “Number1” . You can do this manually by typing cell addresses from the keyboard. Or you can first click inside the field for entering information and then, using the left mouse button pressed, select the required range in the table. If necessary (if you need to mark cells and ranges of cells elsewhere in the table), proceed to filling in argument “Number2” etc. When ready, click OK .
- Get the result in the selected cell.
- The average value may not always be “beautiful” due to the large number of decimal places. If we do not need such detailing, it can always be configured. To do this, right-click on the resulting cell. In the context menu that opens, select item “Cell Format” .
- Being in the tab “Number” select format “Numerical” and on the right side of the window indicate the number of decimal places after the decimal point. In most cases, two digits is more than enough. Also, when working with large numbers, you can check the “Digit group separator” box. After making changes, press the button OK .
- Everything is ready. Now the result looks much more attractive.
Tools in the Formulas tab
Excel has a special tab for working with formulas. In the case of calculating the average value, it can also come in handy.
- We stand in the cell in which we plan to perform calculations. Switch to tab “Formulas”. In the tool section “Function Library” click on the icon “Other functions” , in the drop-down list select the group “Statistical” , then — “AVERAGE” .
- The familiar argument window for the selected function opens. We fill in the data and press the button OK .
Manual entry of a function into cell
Like all other functions, the formula AVERAGE with the necessary arguments can be immediately entered in the desired cell.
In general, the syntax of the function AVERAGE looks like this:
=AVERAGE(number1;number2;...) numeric values.
= AVERAGE(3; 5; 22; 31; 75)
Here's what it looks like with cell references in our case. Let's say we decide to include in the count the entire first row and only three values from the second:
=AVERAGE(B2;C2;D2;E2;F2;G2;h3;B3;C3;D3)
When the formula is complete, press Enter and get the finished result.
Of course, this method cannot be called convenient, but sometimes, with a small amount of data, it may well be used.
Determining the average value by condition
0002 In addition to the methods listed above, Excel also provides the ability to calculate the average value according to a user-specified condition. As follows from the description, only numbers (cells with numeric data) that meet a specific condition will participate in the total count.
Suppose we need to calculate the average value only for positive numbers, i.e. those greater than zero. In this case, the function AVERAGEIF will help us out.
- We stand in the resulting cell and press the button "Insert function" (fx) to the left of the formula bar.
- In the Function Wizard select category “Statistical” , click on operator next to “AVERAGEIF” and press OK .
- The arguments of the function will open, after filling which we click OK:
- in the value of the argument “Range ” we indicate (manually or by selecting with the left mouse button in the table itself) the required area of cells;
- in the value of the argument “Condition” , respectively, we set our condition for cells from the marked range to get into the general calculation. In our case, this expression is “>0” . Instead of a specific number, if necessary, you can specify the address of a cell containing a numeric value in the condition.
- Argument field “Average_range” can be left empty, since its mandatory filling is required only when working with text data.
- The average value, taking into account the cell selection condition we specified, was displayed in the torn cell.
Conclusion
Thus, in Excel there are many ways to find the average value both for individual rows and columns, and for entire ranges of cells, which, moreover, can be scattered around the table. And the use of a particular method is determined by the convenience and expediency of its use in each specific case.
What is the purpose of the correlation search? Why is it used if correlation does not imply causation?
Have you ever come across strange statistics about two events that at first glance seem unrelated? For example, if you ask a person to predict sales of air conditioners based only on sales of ice cream, the prediction may seem ridiculous. After all, air conditioners and ice cream are two different consumer goods produced by unrelated industries. It can be argued that ice cream has as much to do with air conditioning as planet Earth has to do with Hale's comet.
Scatterplot of two random variables, X and Y. It can be seen that an increase in X correlates with an increase in Y. But it remains to be established that an increase in X causes an increase in Y.
Or, for example: this study in which temperature correlated with transmission of COVID-19. However, this study, based on data from the same time period in which the opposite conclusion was made, did not receive the same attention. Why is it so? Does this mean that the search for correlation is useless?
Or, for example: In the first half of 2020, the media was overwhelmed with information about this study related to the correlation of temperature with the transmission of COVID-19. However, this study, conducted on data available over the same time interval, in which a different conclusion was drawn, did not receive the same attention. Why is it so? Does this mean that correlation detection is useless? No.
Let's first understand what a correlation is before we start looking for its merits. We then move on to causation.
Finding Meaning in Random Data: Exploratory Analysis
An event is any phenomenon that can be observed (may be written as a number). For example, sales of air conditioners, grades received by students in the class, goals scored by a player, etc. These random real-life events are stored as data from which salespeople, teachers, and coaches can draw certain conclusions.
When many data points (numeric values) are available for a random event, the event is called a random variable (random because the values they take cannot be predicted before the event occurs, and variable because the values change with each new occurrence).
When considering two random variables, it may turn out that there is some relationship between them, which will help to better understand the events and make accurate predictions about the future outcome of these events. This is very convenient when there is a limited amount of initial data.
Two basic statistical concepts need to be introduced to help us understand correlation better.
The first is dispersion. If a random variable X has n data points, the variance describes the mean difference of each data point from the mean value of X. When plotted, the variance shows the spread of the values. A more scattered dataset will have higher variance than a closely spaced dataset.
Variance indicates the spread of data points about the mean. The variance of the green variable is greater than the variance of the red variable.
Second - covariance. Given two random variables X and Y, a change in the values of one variable may or may not be associated with a change in the values of the other variable. Covariance assigns a numeric value to this value trend.
Correlation
Correlation is a mathematical tool used to identify a relationship between two random events. The goal is to find out the degree of proximity of disparate points to a straight line (linear relationship). Given n data points on two events X and Y, the correlation, r, is defined as follows:
where,
cov(X, Y) = covariance between X and Y
What is the purpose of looking for correlation? Why is it used if correlation does not imply causation?,
X and Y variance respectively
From the mathematical definition of correlation, always .
The following cases occur:
If r=1, then the data points lie on a straight line and there is no scattering. We say that X is linearly correlated with Y. This means that a change in X results in a proportional change in Y, which on the graph is a straight line with a positive slope.
If r=-1, then the data points also lie on a straight line and there is no scattering. X is still linearly correlated with Y. But a change in X results in an inversely proportional change in Y, which is a straight line with a negative slope on the graph.
If -1 < r < 0, then the points remain scattered around the best fitting line with a negative slope.
If 0 < r < 1, then the points remain scattered around the approximating straight line with a positive slope.
Correlation coefficient and related graphs.
Correlation cannot imply causality
Having learned the basics of correlation, let's delve into its interpretation. Often, quite erroneously, the correlation between two random variables X and Y is interpreted as a causal relationship, i.e. X calls Y. Take the example of air conditioner sales (X) and ice cream sold (Y). If a positive correlation is found (say r = 0.8), does that mean that X caused Y, or vice versa? No. This means that there is probably some other factor (Z) that is common to both X and Y. What could Z be?
What random variable would cause a positive change in sales of air conditioners and ice cream? That random variable could probably be temperature.
Think about it. Ice cream is a dessert that is much more likely to be consumed in summer than in other seasons. Rising temperatures may well lead more people to buy these desserts to cool off.