## Math 10 Chapter 5 Lesson 3: Average, median, mode

## 1. Summary of theory

### 1.1. Average (or average)

Given a statistics table (values) of a sign \(x\). The ratio of the sum of all table values to the number of table values is the average, denoted by \(\overline{x}\). The formula for calculating the average is as follows:

**a) For the discrete frequency distribution table**

\(\overline{x} = \frac{1}{n}.({n_1}{x_{1}} + {\rm{ }}{n_2}{x_2} + \ldots + {\rm{ }} {n_n}{x_n}){\rm{ }} \)

\(= {\rm{ }}{f_1}{x_{1}} + {\rm{ }}{f_2}{x_{2}} + \ldots + {\rm{ }}{f_n}{x_n} .\) (first)

Where \({n_i},{\rm{ }}{f_{i}}\left( {i = {\rm{ }}1,{\rm{ }}2, \ldots ,{\rm{ } }k} \right)\) is the frequency, respectively, the frequency of the value \(x_i, n\) is the number of statistics with \(n_1+ n_2+…+ n_n= n\).

**Note:** The formula (1) also has a short form as follows:

\(\overline{x}=\frac{1}{n}\sum_{i=1}^{k}n_{i}x_{i}=\sum_{i=1}^{k}f_{i }x_{i}\)

**b) For the frequency distribution table, we have:**

\(\overline{x} = \frac{1}{n}.({n_1}{C_{1}} + {\rm{ }}{n_2}{C_{2}} + \ldots + {\rm { }}{n_k}{C_k}){\rm{ }}\)\( = {\rm{ }}{f_1}{C_{1}} + {\rm{ }}{f_2}{C_{2 }} + \ldots + {\rm{ }}{f_k}{C_k}\)

Where \(({n_i},{\rm{ }}{C_i},{\rm{ }}{f_i}\) is the frequency, representation value, frequency of the \(i)th class, respectively. (i = 1, 2, …, k)\).

### 1.2. Median

**– Concept:** When statistics have large variances, the average is not representative of the data. Then we choose another number of features that are more appropriate, that is the median

– Sort the statistical values in non-decreasing order.

- If there are \(n\) numbers, \(n\) odd \((n = 2k + 1)\) then \({M_e} = {x_{k + 1}}\) is called the median.
- If \(n\) is an even number \((n = 2k)\), then the median is \(M_{e}=\frac{x_{k}+x_{k+1}}{2}. \)

**– Note how to find the median:**

- Must arrange the order of statistics into a non-decreasing (or non-increasing) sequence.
- If n is odd, then Me is the middle number (the \(\frac{{n + 1}}{2}\)).
- If n is even, then Me is the average of the two middle numbers (the \(\frac{n}{2}\) and the \(\frac{n}{2} + 1\)))) .

### 1.3. Fashion

**– Define:** The mode of a frequency distribution table is the value with the highest frequency and is denoted Mo.

**– Comment:** A data sample can have one or more modes.

## 2. Illustrated exercise

**Question 1:** Given the following table of frequency distribution and frequency of layering:

Average temperature of December in Vinh city from 1961 to the end of 1990 (30 years)

Calculate the average of tables 6 and 8.

**Solution guide**

The average of table 6 is:

\(\overline x = {{16,7} \over {100}}.16 + {{43.3} \over {100}}.18 + {{36,7} \over {100}}.20 + {{3,3} \over {100}}.21\)\(\, \approx 18.53\)

The average of table 8 is:

\(\overline x = \dfrac{{13.1 + 15.2 + 17.12 + 19.9 + 21.5}}{{30}} \)\(\,\approx 17.93\)

**Verse 2:** In the frequency distribution table, the statistics have been ordered in a non-decreasing sequence according to their values. Find the median of the following quarter-per-quarter shirt sales statistics at a men’s shirt store:

**Solution guide**

We sort the number of shirts sold in ascending order:

36, 36, 36, …, 36, 37, 37, …, 37, 38, 38, …, 38, …., 42, 42.

The sequence consists of 465 numbers, so the median is the number in the 233rd place.

Number 233 is number 39.

The median of the above data is: M_{e} = 39

## 3. Practice

### 3.1. Essay exercises

**Question 1:** In a high school, in order to find out the math learning situation of two classes 10A and 10B, those two classes were given the same math test at the same time and created the following two frequency distribution tables:

Test scores of class 10A:

Test scores of class 10B:

Calculate the averages \(\overline x ,\overline y \) of the two distribution tables above and comment on the test results of the two classes.

**Verse 2: **The monthly salary of 7 employees in a travel agency is as follows (in thousands of VND): 650, 840, 690, 720, 2500, 670, 3000.

Find the median of the given statistics. State the meaning of the results found.

### 3.2. Multiple choice exercises

**Question 1: **Three groups of students consisted of 20 students, 15 students, and 25 students. The average weight of each group was 50kg, 38kg, and 40kg, respectively. The average weight of all three groups of students was

A. 41.6 kg

B. 42.8 kg

C. 41.8 kg

D. Another answer

**Verse 2: **For the series of statistics: 48,36,33,38,32,48,42,33,39. Then the median is

A. 32

B. 36

C. 38

D. 40

**Question 3: **Given the statistical sample \(\left\{ {6,5,5,2,9,10.8} \right\}\) .What is the mode of the above data sample?

A. 5

B. 10

C. 2

D. 6

**Question 4: **Given the statistical sample \(\left\{ {28,16,13,18,12,28,13,19} \right\}\) .What is the median of the above data sample?

A. 14

B. 16

C. 17

D. 18

**Question 5: **A student’s semester exam score is as follows: 4; 6; 2; 7; 3; 5; 9; 8; 7; 10; 9. The mean and median are , respectively

A. 6,22 and 7

B. 7 and 6

C. 6.4 and 7

D. 6 and 6

**Question 6: **Given the statistical sample \(\left\{ {8,10,12,14,16} \right\}\).The mean of the above data sample is

A. 12

B. 14

C. 13

D. 12.5

## 4. Conclusion

Through this lesson, you should achieve the following goals:

- Understand the most basic properties of Mean, Median, Mode.
- Solve related exercises.

.

=============