In math what does average mean: Definition of Average
Posted onMean vs Median — Difference and Comparison
Definitions of mean and median
In mathematics and statistics, the mean or the arithmetic mean of a list of numbers is the sum of the entire list divided by the number of items in the list. When looking at symmetric distributions, the mean is probably the best measure to arrive at central tendency. In probability theory and statistics, a median is that number separating the higher half of a sample, a population, or a probability distribution, from the lower half.
How to calculate
The Mean or average is probably the most commonly used method of describing central tendency. A mean is computed by adding up all the values and dividing that score by the number of values. The arithmetic mean of a sample is the sum the sampled values divided by the number of items in the sample:
The Median is the number found at the exact middle of the set of values. A median can be computed by listing all numbers in ascending order and then locating the number in the center of that distribution. This is applicable to an odd number list; in case of an even number of observations, there is no single middle value, so it is a usual practice to take the mean of the two middle values.
Example
Let us say that there are nine students in a class with the following scores on a test: 2, 4, 5, 7, 8, 10, 12, 13, 83. In this case the average score (or the mean) is the sum of all the scores divided by nine. This works out to 144/9 = 16. Note that even though 16 is the arithmetic average, it is distorted by the unusually high score of 83 compared to other scores. Almost all of the students’ scores are below the average. Therefore, in this case the mean is not a good representative of the central tendency of this sample.
The median, on the other hand, is the value which is such that half the scores are above it and half the scores below. So in this example, the median is 8. There are four scores below and four above the value 8. So 8 represents the mid point or the central tendency of the sample.
Comparison of mean, median and mode of two lognormal distributions with different skewness.
Disadvantages of Arithmetic Means and Medians
Mean is not a robust statistic tool since it cannot be applied to all distributions but is easily the most widely used statistic tool to derive the central tendency. The reason that mean cannot be applied to all distributions is because it gets unduly impacted by values in the sample that are too small to too large.
The disadvantage of median is that it is difficult to handle theoretically. There is no easy mathematical formula to calculate the median.
Other Types of Means
There are many ways to determine the central tendency, or average, of a set of values. The mean discussed above is technically the arithmetic mean, and is the most commonly used statistic for average. There are other types of means:
Geometric Mean
The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers x_{1},x_{2},…,x_{n}, the geometric mean is defined as
Geometric means are better than arithmetic means for describing proportional growth. For example, a good application for geometric mean is calculating the compounded annual growth rate (CAGR).
Harmonic Mean
The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. The harmonic mean H of the positive real numbers
x_{1},x_{2},…,x_{n} is
A good application for harmonic means is when averaging multiples. For exampe, it is better to use weighted harmonic mean when calculating the average price–earnings ratio (P/E). If P/E ratios are averaged using a weighted arithmetic mean, high data points get unduly greater weights than low data points.
Pythagorean Means
The arithmetic mean, geometric mean and harmonic mean together form a set of means called the Pythagorean means. For any set of numbers, the harmonic mean is always the smallest of all Pythagorean means, and the arithmetic mean is always the largest of the 3 means. i.e. Harmonic mean ≤ Geometric mean ≤ Arithmetic mean.
Other meanings of the words
Mean can be used as a figure of speech and holds a literary reference. It is also used to imply poor or not being great. Median, in a geometric reference, is a straight line passing from a point in the triangle to the centre of the opposite side.
References
 wikipedia:Mean
 wikipedia:Median
 Modes, Medians and Means: A Unifying Perspective
 Pythagorean means
Mean, Median and Mode
We use statistics such as the mean, median and mode to obtain information about a population from our sample set of observed values.
Mean
The mean (or average) of a set of data values is the sum of all of
the data values divided by the number of data values. That is:
Example 1
The marks of seven students in a mathematics test with a maximum possible
mark of 20 are given below:
15
13 18
16 14
17 12
Find the mean of this set of data values.
Solution:
So, the mean mark is 15.
Symbolically, we can set out the solution as follows:
So, the mean mark is 15.
Median
The median of a set of data values is the middle value of the data
set when it has been arranged in ascending order. That is, from the
smallest value to the highest value.
Example 2
The marks of nine students in a geography test that had a maximum
possible mark of 50 are given below:
47
35 37
32 38
39 36
34 35
Find the median of this set of data values.
Solution:
Arrange the data values in order from the lowest value to the highest
value:
32
34 35
35 36
37 38
39 47
The fifth data value, 36, is the middle value in this arrangement.
Note:
In general:
If the number of values in the data set is even, then the median is
the average of the two middle values.
Example 3
Find the median of the following data set:
12
18 16
21 10
13 17 19
Solution:
Arrange the data values in order from the lowest value to the highest
value:
10
12 13
16 17
18 19 21
The number of values in the data set is 8, which is even. So, the
median is the average of the two middle values.
Alternative way:
There are 8 values in the data set.
The fourth and fifth scores, 16 and 17, are in the middle. That is,
there is no one middle value.
Note:
 Half of the values in the data set lie below the median
and half lie above the median.  The median is the most commonly quoted figure used to
measure property prices. The use of the median avoids the problem of
the mean property price which is affected by a few expensive properties that
are not representative of the general property market.
Mode
The mode of a set of data values is the value(s) that occurs most
often.
The mode has applications in printing. For example, it is important
to print more of the most popular books; because printing different books in
equal numbers would cause a shortage of some books and an oversupply of
others.
Likewise, the mode has applications in manufacturing. For example, it is
important to manufacture more of the most popular shoes; because
manufacturing different shoes in equal numbers would cause a shortage of
some shoes and an oversupply of others.
Example 4
Find the mode of the following data set:
48 44
48 45 42
49 48
Solution:
The mode is 48 since it occurs most often.
Note:
 It is possible for a set of data values to have more than one
mode.  If there are two data values that occur most frequently, we
say that the set of data values is bimodal.  If there is no data value or data values that occur most
frequently, we say that the set of data values has no mode.
The mean, median and mode of a data set are collectively known as measures
of central tendency as these three measures focus on where the data is
centred or clustered. To analyse data using the mean, median and mode, we
need to use the most appropriate measure of central tendency. The following
points should be remembered:
 The mean is useful for predicting future results when there are no
extreme values in the data set. However, the impact of extreme values on
the mean may be important and should be considered. E.g. The impact of a
stock market crash on average investment returns.  The median may be more useful than the mean when there are extreme
values in the data set as it is not affected by the extreme values.  The mode is useful when the most common item, characteristic or value
of a data set is required.
Key Terms
statistics, mean, average, median, mode, measures
of central tendency
simple explanation and tasks to prepare for DPA and ZNO
 What is the arithmetic mean?
 How to find the arithmetic mean?
 Arithmetic mean tasks for grade 5
 Arithmetic mean: preparation for ZNO
Students study the definition and methods of finding the arithmetic mean in the 5th grade, but this is also important during the exams in the senior classes. Therefore, we propose to learn or repeat important knowledge, to practice solving problems on the arithmetic mean for grade 5 and more complex ones that can be encountered at the VNO.
What is the arithmetic mean?
In school and everyday life, the phrases “average salary”, “average grade”, “average age” are often used. The basis for all these concepts is the mathematical term arithmetic mean.
The arithmetic mean of several numbers is the fraction of dividing the sum of these numbers by their number. That is, to find it, you need to divide the sum of the numbers by their number.
The formula for the arithmetic mean of two numbers looks like this:
See also: Properties and formulas of logarithms
How to find the arithmetic mean?
Determining the arithmetic mean sounds scary and incomprehensible, but in fact, only two arithmetic operations are performed to find it — addition and division. For example, a student takes three numbers, adds them and divides by 3. The answer will be the arithmetic mean.
Consider an example. You need to find the arithmetic average of the numbers 7 and 3. The first step is you add them and get 10. Then this sum must be divided by 2, that is, by the number of numbers.
 7 + 3 = 10.
 10 : 2 = 5.
The answer of the second step is the arithmetic mean.
Consider another example that is close to all schoolchildren. The student at the end of the quarter is interested in the question of what grade he will have in a particular subject. To get an answer, you need to calculate the average score.
For example, a student received the following points for a quarter: 5, 4, 5, 3, 3, 5, 4, 5, 4, 5. To calculate the average score, you need to add all the marks, and then divide by their number ( 10).
 5 + 4 + 5 + 3 + 3 + 5 + 4 + 5 + 4 + 5 = 43
 43 : 10 = 4. 3
Answer: 4.3 is the arithmetic mean.
See also: Putting together a school backpack: tips and checklist rhymetic, it is important for him to learn how to solve. tasks on this topic, in which not everything is so simple. As with any math problem, you need:
 Carefully study the condition and make a short note in which to indicate all the relationships between numbers;
 Read the question several times to understand exactly what you need to find.
It often happens that a student knows the formula for finding the arithmetic mean, but inadvertently adds the wrong numbers or takes into account extra indicators and makes a mistake in the calculations.
Let’s consider interesting problems on the arithmetic mean, which students solve in the 5th grade.
Problem No. 1
The children worked in the garden and picked pears. Artem picked 2 pears, Emilia — 4, Vova — 6. They put them in a basket and took them home. Mom divided the pears equally between the kids. How many pears did each child get?
To solve the problem, you need to find the arithmetic mean of three numbers.
2+4+6=12 pears (children collected in total)
When the student knows how many fruits were found in total, it is necessary to divide this amount by the number of children.
12:3 = 4 pears (each child received)
Answer: 4 , wrote two independent works on 8 and 9 points, and the control — 9. What will be its thematic assessment?
Thematic score is close to the average score for all types of work. To answer the main question of the problem, you must first find the arithmetic mean of all estimates. To do this, add all the points received.
7 + 12 + 8 + 9 + 9 = 44
Further, the total number of points must be divided by the number of grades, that is, by 5.
44 : 5 = 8 .8
8 .8 is the arithmetic mean, but there is no such score, so the teacher will round this number up to 9.
Answer: 9 points is the student’s thematic score.
Problem No. 3
The arithmetic mean of 4 numbers is 3.4, and the arithmetic mean of the remaining 6 numbers is 8.3. Find the arithmetic mean of all 10 numbers.
In such problems, students often make mistakes and find the arithmetic mean of two numbers — 3.4 and 8.3. But it is important to consider that there are not 2 numbers here, but 10. Therefore, in order to find the arithmetic mean of 10 numbers, you need to find out the sum of these 10 numbers.
First we find the sum of 4 numbers. To do this, the arithmetic mean must be multiplied by the number of numbers.
3.4 * 4 = 13.6
Now we calculate the sum of 6 numbers in the same way.
8.3 * 6 = 49.8
The next step is to find the sum of 10 numbers.
13.6 + 49.8 = 63.4
It remains to find the arithmetic mean of the numbers.
63.4 : 10 = 6.34
Answer: 6.34 is the arithmetic mean of 10 numbers.
Problem No. 4
The average age of the football team players who participated in the game is 22 years. After a player was sent off for a foul, the average age of the remaining players was 21. How old was the player who was kicked off the field?
First, the student must find the sum of the years of all the players who have participated in the game since the first minute. To do this, remember that a football team is 11 people.
22*11 = 242(l)
Now you need to find the sum of the years of the players remaining on the field after the removal of one player, that is, 10 people.
21*10 = 210(l)
At the last stage, the student finds the difference between two arithmetic means and finds out the age of the player who broke the rules and was disqualified.
242 — 210 = 32 (l)
Answer: 32 years.
Task No. 5
A farmer harvested 30.2 tons of wheat from each hectare of a field with a total area of 30 hectares, and 32.3 tons of wheat from each hectare of a field of 20 hectares. What is the average yield per hectare for a farmer?
In order not to get confused in the data, it is important for the student to make a schematic record of the condition.
30 ha — 30.2 tons per 1 ha
20 ha — 32.3 tons per 1 ha
It is necessary to find the entire crop from both fields.
 30 * 30.2 = 906 (t) collected from a field with an area of 30 hectares.
 20*32.3=646(t) was collected from a field with an area of 20 ha.
 906+646=1522(t) collected from two fields.
Now the student must determine the total area of the fields and divide the entire crop into it.
 30 + 20 = 50 (ha) area of two fields.
 1522 : 50 = 31.04 (t)
Answer: 31.04 tons is the average yield of wheat per hectare of field.
Task No. 6
The car drove 3.4 hours on the highway at a speed of 90 km/h and 1.6 hours on a dirt road. At what speed was the car traveling on a dirt road if the average speed for the entire journey was 75.6 km / h.
In this problem, the average speed is known and you need to find the speed on one part of the path, that is, this is an inverse problem for the average speed.
3.4 year — 90 km/h
1.6 year — x km/h
avg. speed — 75.6 km / h
We make an equation based on the arithmetic mean formula and simplify it.
(3.4 * 90 + 1.6 hours): (3.4 + 1.6) = 75.6
(306 + 1.6 hours): 5 = 75.6
306 + 1.6 h = 378
1.6 h = 378 — 306
1. 6 h = 72
x \u003d 72: 1.6
x \u003d 45
Answer: 45 km / h is the speed at which the car was moving on a dirt road.
In order to learn how to correctly and quickly solve problems from this topic, it is important for a student to practice a lot and have highquality knowledge in other topics of mathematics. When a child has poorly learned division, he will not be able to find the arithmetic mean of numbers. And a poor understanding of the arithmetic mean will not allow you to solve problems at an average speed.
If a student has problems with discipline, it is important to seek help from a math tutor in time. Incomprehensible topics will accumulate like a snowball, and school performance and motivation to learn will decline.
The teacher will conduct a test in which he will find gaps in knowledge, communicate with the student and learn about his problems and goals for studying mathematics, based on the information received, he will draw up an individual learning plan.
During the lessons, the teacher focuses on only one student and conducts the lesson at a pace that is comfortable for him, devoting the necessary time to each question. You can find a tutor in mathematics for current lessons or preparation for UPE on the BUKI website.
Arithmetic mean: preparation for the ZNO
In the tasks of the main session of the ZNO, there may be a task to find the arithmetic mean. Therefore, when preparing, it is important to repeat this topic and consider examples of similar tasks from past years.
A student from Monday to Friday recorded the time in minutes that he spent on the way to and from school.
Monday 
Tuesday 
Wednesday 
Thursday 
Friday 

To school 
19 
20 
21 
17 
23 
From school 
28 
22 
20 
25 
30 
Find how many minutes on average the road from school was longer than the road to school. To answer the question, you must first find the average travel time to school and the average travel time from school.
To find the average travel time to school, add all the minutes and and divide by the number of days.
(19 + 20 + 21 + 17 + 23): 5 = 100 : 5 = 20 (min.)
Average travel time from school is calculated similarly.
(28 + 22 + 20 + 25 + 30): 5 = 125 : 5 = 25 (min) average travel time from school.
Now we need to find the difference between these average times.
25 – 20 = 5 (min)
Answer: The road from school lasted on average 5 minutes longer than the road to school.
In order to prepare well for UPE, it is important to study systematically and be able to get an explanation of all incomprehensible topics from an experienced teacher. You can find a tutor in mathematics or another discipline on the BUKI website.
See also: Geometric progression: explanation and formulas
Fundamentals of mathematical statistics — what is it, definition and answer
Working with basic statistical terms is usually done with some kind of data. For example, students were weighed in the class. The weight of each one is the weighing data. For convenience, such data is written out in a row, listing all the results obtained. It will be called number series .
It is the numerical series that is analyzed using statistical terms and methods.
RANGE AND FREQUENCY:
Such a characteristic as a range shows how much the elements of a series differ from each other.
Imagine a number series consisting of natural numbers:
3, 11, 18, 18, 22, 37, 37, 37, 89
The span of is the difference between the largest and smallest numbers in the series.
In this case, the largest number is 89, and the smallest is 3.
3, 11, 18, 18, 22, 37, 37, 37, 89
Then the range of this series is \(893 = 86\).
Frequency is the number of repetitions of the same element.
For example, the frequency of the number 3 is 1 time. But the number 18 occurs 2 times.
FASHION AND MEDIAN:
To understand what averages and popular values are present in the series, such characteristics of the number series as mode and median are used.
The mode is the number of the series that occurs the most times.
In this case, the number is 37. It occurs three times. More than it, no other number occurs. So it is fashion.
3, 11, 18, 18, 22, 37, 37, 37, 89
The median of is the element of the series, standing exactly in the middle.
For example, in our series this number is 22.
3, 11, 18, 18, 22, 37, 37, 37, 89
greater than 22.
Therefore, it is important to arrange the elements in order in order to immediately notice the median.
If we add one more number to our series, for example, add the number 15, we get 10 numbers in the series. Then in the middle of the row there will already be two numbers: 22 and 18:
3, 11, 15, 18, 18, 22, 37, 37, 37, 89
In this case, if there is an odd number of elements in a row, its median will be , the arithmetic mean of these elements.
ARITHMETIC AVERAGE:
The arithmetic mean of a series is a number by which each element of the series can be replaced so that their sum does not change.
It is found by the following formula:
This entry means that in order to find the arithmetic mean of a series of numbers, you need to add all its elements and divide by their number.
Find the arithmetic mean of the fifth and sixth element of our series. It will be the median, because both of these elements are in the middle of the row:
3, 11, 15, 18, 18, 22, 37, 37, 37, 89
\(m = \frac{18 + 22}{2} = \frac{40 }{2} = 20\)
Indeed, if we replace the sum of these elements by the sum of the same number of arithmetic means, then this sum will not change.
\(18 + 22 \u003d 20 + 20\)
Find the arithmetic mean of the entire series. To do this, add all the elements and divide already by their number:
\(\overline{a_{x}} = \frac{3 + 11 + 15 + 18 + 18 + 22 + 37 + 37 + 37 + 89}{10} = \frac{287}{10} = 28 ,7 \)
GEOMETRIC AVERAGE:
The geometric mean is a number by which all elements of the series can be replaced so that their product does not change.
It is found by the formula:
\(\overline{g_{x}} = \sqrt[n]{x_{1}x_{2}x_{3} \bullet … \bullet x_{n}} \)
This means that to find the geometric mean, you need to multiply all the elements of the series and extract the product from the root, the degree of which is equal to their number.
Take a row with fewer elements:
1, 3, 3, 5, 8
Find its geometric mean:
\(\overline{g_{x}} = \sqrt[5]{1 \bullet 3 \bullet 3 \bullet 5 \bullet 8} = \sqrt[5]{360} \approx 3.25\)
Example #1:
At the end of the year, it is important for a student to get a good quarter grade. He has the following grades in mathematics:
4, 3, 5, 5, 4, 4, 2, 4, 5, 5, 3, 4
Let’s analyze his grades completely using statistical characteristics.
let’s put all its elements in order:
2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5
2. Determine the range of the series. The student received all possible grades for the quarter, and as we can see, the highest grade is 5, and the smallest grade is 2. Then the range is 5 – 2 = 3.
3. Let’s determine the mode of the series. The most common rating is rating 4. Its frequency is 5.
4. At the same time, the frequency of rating 5 is slightly less and equal to four.
5. The median of this series will be the number in the middle. In total, we have 12 elements of the series, then the sixth and seventh elements will be the average values.
2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5
4}{2} = 4\)
Because the elements are equal, their arithmetic mean will be the same. The median of this series is 4
6. The arithmetic mean of the series is
\(\overline{a_{x}} = \frac{2 + 3 + 3 + 4 + 4 + 4 + 4 + 4 + 5 + 5 + 5 + 5}{12} = \frac{48}{ 12} = 4\)
7. Geometric mean of the series:
\(\overline{g_{x}} = \sqrt[12]{2 \bullet 3 \bullet 3 \bullet 4 \bullet 4 \bullet 4 \ bullet 4 \bullet 4 \bullet 5 \bullet 5 \bullet 5 \bullet 5} = \sqrt[12]{11520000} \approx 3.88\)
Thus, we have determined that for a quarter the student will receive the arithmetic mean of all grades , i.