Question

Consider a sample with data values of 27, 25, 20, 15, 30, 34, 28, and 25. Compute the range, interquartile range, variance, and standard deviation (to a maximum of 2 decimals, if decimals are necessary).
Range:
Interquartile range:
Variance:
Standard deviation:

Answer 1

Answer:

Range: 19

Interquartile range: 6.5

Variance: 44.86

Standard deviation: 6.70

Step-by-step explanation:

Given set:

27, 25, 20, 15, 30, 34, 28, 25.

(i) Range is the difference between the largest value and smallest value in a given data.

In the data above, the;

largest value = 34

smallest value = 15

Range = largest - smallest

Range = 34 - 15

Range = 19

(ii) Interquartile range is the difference between the median (Q₁) of the lower half of the given data, and the median (Q₃) of the upper half of the given data. i.e Q₃ - Q₁.

This should be done after ordering the given data in ascending order.

Ordering the above data gives:

15, 20, 25, 25, 27, 28, 30, 34

In the data above;

lower half = 15, 20, 25, 25

upper half = 27, 28, 30, 34

Find median (Q₁)  of the lower half

Since there are even number of data, the median is the average of two middle numbers (20 and 25) i.e.

Q₁ = $\frac{20+25}{2}$ = $\frac{45}{2}$ = 22.5

Find median (Q₃)  of the upper half

Since there are even number of data, the median is the average of two middle numbers (28 and 30) i.e.

Q₃ = $\frac{28+30}{2}$ = $\frac{58}{2}$ = 29

Interquartile range = Q₃ - Q₁

Interquartile range = 29 - 22.5

Interquartile range = 6.5

(iii) Variance is the measure of variability of a given set of data. It is calculated by following these steps:

=> First find the mean of the data by summing the numbers and dividing by the number of numbers. i.e

mean(x) = [15 + 20 + 25 + 25 + 27 + 28 + 30 + 34] / 8

mean(x) = [204] / 8

mean(x) = 25.5

=> Next, get the mean deviation by subtracting the mean from each individual value in the given set of data. i.e

15 - 22.5 = -7.5

20 - 22.5 = -2.5

25 - 22.5 = 2.5

25 - 22.5 = 2.5

27 - 22.5 = 4.5

28 - 22.5 = 5.5

30 - 22.5 = 7.5

34 - 22.5 = 11.5

=> Next, find the square of each of the results obtained above. i.e

(-7.5)² = 56.25

(-2.5)² = 6.25

(2.5)² = 6.25

(2.5)² = 6.25

(4.5)² = 20.25

(5.5)² = 30.25

(7.5)² = 56.25

(11.5)² = 132.25

=> Next, add all the results above together as follows;

56.25 + 6.25 + 6.25 + 6.25 + 20.25 + 30.25 + 56.25 + 132.25 = 314

=> Now, obtain the variance by dividing the result above one less than the number of numbers.

The number of numbers is 8. One less than this gives 7.

Therefore,

Variance = 314 / 7

Variance = 44.86

(iv) Standard deviation is the measure of how the numbers in the given data are spread out. It is calculated as the square root of variance.

Therefore,

Standard deviation = √variance.

Variance has been calculated above.

Standard deviation = √44.86

Standard deviation = 6.70