Question

Suppose you have a data set with 9 distinct observations. if the largest number in the data set increases the standard deviation will:___________

Answer 1

If the largest number in the data set increases the standard deviation will increase.

Let us suppose that we are given a data set with 9 distinct observations.

It is given that the largest number in the data set increases.

We know that,

Standard deviation = $\sqrt{ \frac{sum[(X-m)^{2}]}{N}}$

Where,

X = Element of data set

m = mean of the data set

N = number of elements in the data set

Let us consider our data set to be:-

1,2,3,4,5,6,7,8,9

Here,

Mean, m = (1+2+3+4+5+6+7+8+9)/9 = 45/9 = 5

Hence,

$sum(X-m)^{2}=(1-5)^{2}+(2-5)^{2}+(3-5)^{2}+(4-5)^{2}+(5-5)^{0}+(6-5)^{2}+(7-5)^{2}+(8-5)^{2}+(9-5)^{2} = 16+9+4+1+0+1+4+9+16=60$

N = 9

Hence

Standard deviation = $\sqrt{\frac{60}{9} }=\sqrt{\frac{20}{3} } =2.58 (approximately)$

Now, let the largest element of the set that is 9 increases to 18.

Hence,

Mean , m = (1 +2+3+4+5+6+7+8+18)/9 = 54/9 = 6

N = 9

$sum(X-m)^{2}=(1-6)^{2}+(2-6)^{2}+(3-6)^{2}+(4-6)^{2}+(5-6)^{0}+(6-6)^{2}+(7-6)^{2}+(8-6)^{2}+(18-6)^{2} = 25+16+9+4+1+0+1+4+144=206$

Standard deviation = $\sqrt{\frac{206}{9} }=4.78(approximately)$

Hence, we can clearly see that the standard deviation increases.

To learn more about standard deviation, here:-

https://brainly.com/question/13905583

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