In math what does average mean: Definition of Average

• 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.

1. 30 * 30.2 = 906 (t) collected from a field with an area of ​​30 hectares.
2. 20*32.3=646(t) was collected from a field with an area of ​​20 ha.
3. 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.

1. 30 + 20 = 50 (ha) area of ​​two fields.
2. 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 high-quality 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.

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 \(89-3 = 86\).

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.