Standard Deviation: Approximately 7.47 Interpretation: See solution.
Practice makes perfect
We want to find the standard deviation of the given data set.
25 , 20 , 22 , 27 , 48 , 32
19 , 27 , 25 , 22 , 21 , 24
To use the formula for the standard deviation, we need to find the mean first.
Mean
The mean of a data set, usually denoted by x, is calculated by finding the sum of all values and then dividing by the number of values in the set. In this case, there are 12 values in the set.
The standard deviation of a set of data is the average amount by which each individual value deviates or differs from the mean.
Standard Deviation
sqrt((x_1-x )^2 + (x_2-x )^2 + ... + (x_n-x )^2/n)
In the above formula, x_1, ... ,x_n are the values of the set of data, x is the mean, and n is the number of values. We have 12 values and the mean is x=26. Let's apply the formula to each value in the set.
x_i
x-x
(x-x)^2
25
25 - 26=- 1
(- 1)^2=1
20
20 - 26=- 6
(- 6)^2=36
22
22 - 26=-4
(- 4)^2=16
27
27 - 26=1
1^2=1
48
48 - 26=22
22^2=484
32
32 - 26=6
6^2=36
19
19 - 26=-7
(- 7)^2=49
27
27 - 26=1
1^2=1
25
25 - 26=- 1
(- 1)^2=1
22
22 - 26=-4
(- 4)^2=16
21
21 - 26=-5
(- 5)^2=25
24
24 - 26=-2
(- 2)^2=4
Sum of Values
670
Finally, we need to divide by 12 and then calculate the square root.
Standard Deviation: sqrt(670/12)≈ 7.47...
We found that standard deviation of the given data set is approximately 7.47. This means that the typical age of a contestant on Show B differs from the mean of 26 years by about 7.47 years.
