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 x̄_{1} and x̄_{2 }be the arithmetic means of two datum and n_{1} and n_{2} be the respective number values in each data.
Combined Mean :
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 :
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.
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
x̄_{1 }---> average test score of 15 boys, x̄_{1} = 83
x̄_{2 }---> average test score of 15 girls, x̄_{2} = ?
n_{1 }---> number of boys, n_{1 }= 15
n_{2 }---> number of girls, n_{2 }= 12
Combined Mean :
Substitute x̄ = 81, x̄_{1} = 83, n_{1 }= 15 and n_{2 }= 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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