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A sample of size n = 10 is drawn from a population. The data is shown below.

115.6
109.3
126
104.9
131.9
113.7
119.8
98.6
131.9
131.9
What is the range of this data set?

What is the standard deviation of this data set? (Remember, it is a sample.) Please report the answer with appropriate rounding, reporting 2 more decimal places than the original data. ​

User Fuzzygroup
by
8.2k points

1 Answer

1 vote

Answer:

first, arrange the numbers from least to greatest order. (Actually this is a longer step, if you also want to find the median. But since you are only asking for range and standard deviation, I won't do that here.) So, just find the lowest number and the highest number. Subtract the lowest from the highest. That is your range.

Your problem:

lowest: 98.6

highest: 131.9

subtract: 131.9 - 98.6 = 33.3 ← this is your range

Now, standard deviation.

Standard deviation is the amount of variety you have in your data sample.

Step 1: Find the mean

Add up all your numbers and divide by how many numbers you have.

You have 10 numbers in your sample.

115.6 + 109.3 + 126 + 104.9 + 131.9 + 113.7 + 119.8 + 98.6 + 131.9 + 131.9

total = 1,183.6

now divide this by 10. (n is the variable used here, so n = 10. This is because you have ten numbers. )

so n = 10

and 1,183.6/10 = 118.3

Now subtract each number by 118.3.

115.6 - 118.3 = -2.7

109.3 - 118.3 = -9

126 - 118.3 = 7.7

104.9 - 118.3 = -13.4

131.9 -118.3 = 13.6

113.7 - 118.3 = -4.6

119.8 - 118.3 = 1.5

98.6 - 118.3 = -19.7

131.9 - 118.3 = 13.6

131.9 - 118.3 = 13.6

now square all these numbers

7.29

81

59.29

179.56

184.96

21.16

0.75

388.09

184.96

184.96

Find the sum of these squares now. (We're almost done!)

sum = 1,292.02

remember our n?

it was n=10

now the formula for this is,

sum of squares ÷ n-1

substitute all this in.

1,292.02 ÷ 9 = 143.55

Remember. This is the VARIANCE. NOT the standard deviation.

The last step to find the standard deviation is, to find the square root of what we got. (143.55)

√143.55

= 11.9812353286 this is the number, but rounded two more decimal places is..

11.98 is the standard deviation.

Hope this helped!

User Rajesh Pandya
by
8.4k points