Question

1. for the data in the following sample: 1, 1, 9, 1

a. find the mean, ss, variance, and standard deviation.
b. now change the score of x = 9 to x = 3, and find the new values for ss, variance, and standard deviation.
c. describe how one extreme score influences the mean and standard deviation.

Explanation:

Step1: Calculate the mean for part a

The mean $\bar{X}=\frac{\sum X}{n}$, where $\sum X=1 + 1+9 + 1=12$ and $n = 4$. So, $\bar{X}=\frac{12}{4}=3$.

Step2: Calculate $SS$ for part a

$SS=\sum(X - \bar{X})^2$. For $X = 1$, $(1 - 3)^2=4$; for the three $1$s, the sum of squared - deviations is $3\times4 = 12$, and for $X = 9$, $(9 - 3)^2=36$. So, $SS=12 + 36=48$.

Step3: Calculate the variance for part a

The variance $s^2=\frac{SS}{n - 1}=\frac{48}{3}=16$.

Step4: Calculate the standard deviation for part a

The standard deviation $s=\sqrt{s^2}=\sqrt{16}=4$.

Step5: Calculate the new mean for part b

When $X = 9$ is changed to $X = 3$, $\sum X=1 + 1+3 + 1=6$, and $n = 4$. So, $\bar{X}=\frac{6}{4}=1.5$.

Step6: Calculate the new $SS$ for part b

For $X = 1$, $(1 - 1.5)^2 = 0.25$, for the three $1$s, the sum of squared - deviations is $3\times0.25=0.75$, and for $X = 3$, $(3 - 1.5)^2 = 2.25$. So, $SS=0.75+2.25 = 3$.

Step7: Calculate the new variance for part b

The new variance $s^2=\frac{SS}{n - 1}=\frac{3}{3}=1$.

Step8: Calculate the new standard deviation for part b

The new standard deviation $s=\sqrt{s^2}=\sqrt{1}=1$.

Step9: Analyze the influence for part c

The extreme score of $X = 9$ in the original data set increased the mean from $1.5$ (the mean without the extreme value) to $3$. It also increased the standard deviation from $1$ (without the extreme value) to $4$. An extreme score can pull the mean in its direction and increase the standard deviation, which measures the spread of the data.

Answer:

a. Mean = 3, $SS = 48$, Variance = 16, Standard deviation = 4
b. New $SS = 3$, New variance = 1, New standard deviation = 1
c. An extreme score can increase the mean and the standard - deviation, pulling the mean in its direction and increasing the spread of the data.