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Over [174]
3 years ago
11

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:
Mathematics
1 answer:
Dima020 [189]3 years ago
3 0
<h2>Answer:</h2>

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 = \frac{20+25}{2} = \frac{45}{2} = 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 = \frac{28+30}{2} = \frac{58}{2} = 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

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