Range: 19
Interquartile range: 6.5
Variance: 44.86
Standard deviation: 6.70
<h2>Step-by-step explanation:</h2>
Given set:
27, 25, 20, 15, 30, 34, 28, 25.
(i) <em>Range</em> is the difference between the largest value and smallest value in a given data.
<em>In the data above, the;</em>
largest value = 34
smallest value = 15
Range = largest - smallest
Range = 34 - 15
Range = 19
(ii) <em>Interquartile range </em>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.
<em>Ordering the above data gives:</em>
15, 20, 25, 25, 27, 28, 30, 34
<em>In the data above;</em>
lower half = 15, 20, 25, 25
upper half = 27, 28, 30, 34
<em>Find median (Q₁) of the lower half</em>
Since there are even number of data, the median is the average of two middle numbers (20 and 25) i.e.
Q₁ = =
= 22.5
<em>Find median (</em>Q₃<em>) of the upper half</em>
Since there are even number of data, the median is the average of two middle numbers (28 and 30) i.e.
Q₃ = =
= 29
Interquartile range = Q₃ - Q₁
Interquartile range = 29 - 22.5
Interquartile range = 6.5
(iii) <em>Variance </em>is the measure of variability of a given set of data. It is calculated by following these steps:
<em>=> First find the mean of the data by summing the numbers and dividing by the number of numbers. i.e</em>
mean(x) = [15 + 20 + 25 + 25 + 27 + 28 + 30 + 34] / 8
mean(x) = [204] / 8
mean(x) = 25.5
=> <em>Next, get the mean deviation by subtracting the mean from each individual value in the given set of data. i.e</em>
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
=> <em>Next, find the square of each of the results obtained above. i.e</em>
(-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
=> <em>Next, add all the results above together as follows;</em>
56.25 + 6.25 + 6.25 + 6.25 + 20.25 + 30.25 + 56.25 + 132.25 = 314
=> <em>Now, obtain the variance by dividing the result above one less than the number of numbers.</em>
The number of numbers is 8. One less than this gives 7.
Therefore,
Variance = 314 / 7
Variance = 44.86
(iv) <em>Standard deviation </em>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.
<em>Variance has been calculated above.</em>
<em />
Standard deviation = √44.86
Standard deviation = 6.70