MEAN MEDIAN AND MODE

Mean

The mean of a data set is the sum of the values in the data set divided by the number of values in the data set. The mean is also called the arithmetic mean.

Let x represent a data. 

x̄ ----> arithmetic mean

Σx ----> sum of all values

n ----> number of values

The sum of the values = mean x number of values.

Σx = nx̄

Let x̄₁ and x̄₂ be the arithmetic means of two datum and n₁ and n₂ be the respective number values in each data.

Combined Mean :

Median

The median of a set of data is the middle value when the values in the data are arranged from least to greatest.

For example, in the set of data {3, 7, 8, 10, 14}, the median is 8.

If there are two middle numbers, the median is the mean of the two numbers.

For example, in the set of data {3, 7, 10, 14}, the median is

= (7 + 10)/2

= 17/2

= 8.5

Formula to find median :

Mode

The mode of a set of data is the number that appears most frequently. Some set of data have more than one mode, and others have no mode.

In the set of data {3, 7, 8, 10, 14}, there is no mode since each number appears only once.

In the set of data {3, 7, 7, 10, 14, 19, 19, 25}, the modes are 7 and 19, since both numbers appear twice.

Solved Problems

Problem 1 :

Find the mean, median and  mode of the data set.

{132, 149, 152, 164, 164, 175}

Solution :

Mean :

= sum of the values/number of values

= (132 + 149 + 152 + 164 + 164 + 175)/6

= 936/6

= 156

Median :

132, 149, 152, 164, 164, 175

The values in the data given are already arranged from least to greatest.

Formula to find median :

= [(n + 1)/2]^th value

Substitute n = 6.

= [(6 + 1)/2]^th value

= (7/2)^th value

= 3.5^th value

= average of 3^rd and 4^th values

= (152 + 164)/2

Mode :

In the data given {132, 149, 152, 164, 164, 175}, the value 164 appears more number of times, that is twice.

So, mode = 164.

Problem 2 :

In a geometry class of 15 boys and 12 girls, the average (arithmetic mean) test score of the class was 81. If the average score of the 15 boys was 83, what was the average score of the 12 girls?

Answer :

Let

x̄ ---> average test score of the class, x̄ = 81

x̄₁ ---> average test score of 15 boys, x̄₁ = 83

x̄₂ ---> average test score of 15 girls, x̄₂ = ?

n₁ ---> number of boys, n₁ = 15

n₂ ---> number of girls, n₂ = 12

Combined Mean :

Substitute x̄ = 81, x̄₁ = 83, n₁ = 15 and n₂ = 12.

The average score of 12 girls is 78.5.

Problem 3 :

The test scores of 8 students are shown in the table above. Let m be the mean of the scores and M be the median of the score. What is the value of M - m?

A) -6

B) 0

C) 3

D) 6

Solution :

Find the mean :

Sum of the scores :

= 1x67 + 3x75 + 2x87 + 2x91

= 67 + 225 + 174 + 182

= 648

Total number of students :

= 1 + 3 + 2 + 2

= 8

Mean :

m = 648/8

m = 81

Find the median :

Given data :

67, 75, 75, 75, 87, 87, 91, 91

The values in the given data are already arranged from least to greatest.

Formula to find median :

= [(n + 1)/2]^th value

Substitute n = 8.

= [(8 + 1)/2]^th value

= (9/2)^th value

= 4.5^th value

= average of 4^th and 5^th values

= (75 + 87)/2

= 81

median, M = 81

M - m = 81 - 81

M - m = 0

The correct answer choice is (B).

Problem 4 :

n, n - 3, 2n - 1, 3n - 4 and 5n + 12

The average (arithmetic mean) of five numbers given above is 8. Which of of the following is true?

A) median = 5, mode = 7

B) median = 5, mode = 5

C) median = 7, mode = 7

D) median = 7, mode = 5

Solution :

Average of five numbers = 8

(n + n - 3 + 2n - 1 + 3n - 4 + 5n + 12)/5 = 8

(12n + 4)/5 = 8

Multiply both sides by 5.

12n + 4 = 40

Subtract 4 from both sides.

12n = 36

Divide both sides by 12.

n = 3

Then,

n - 3 = 0

2n - 1 = 5

3n - 4 = 5

5n + 12 = 27

The given five numbers are 3, 0, 5, 5 and 27.

Arrange the numbers from least to greatest.

0, 3, 5, 5, 27

median = 5

mode = 5

The correct answer choice is (B).

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