Chapter 1.3, Problem 82E

(a)

To determine

To find: The mean and the standard deviation for IQ.

Answer to Problem 82E

Solution: The mean is 108.9231 and the standard deviation is 13.17.

Explanation of Solution

Calculation: Mean for the IQ scores can be calculated as:

$\overline{x}=\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{N}$

where N is the total number of observations and ‘i’ varies from 1 to 78.

$\begin{array}{c}\overline{x}=\frac{111+107+100+\cdots +106}{78}\\ =\frac{8496}{78}\\ =108.9231\end{array}$

Standard deviation for the IQ scores can be calculated as:

$s=\sqrt{\frac{{\left({\displaystyle \sum _i{x}_i-\overline{x}}\right)}^2}{n-1}}$

Substitute the values in the formula above:

$\begin{array}{c}s=\sqrt{\frac{{\left({\displaystyle \sum _i{x}_i-\overline{x}}\right)}^2}{n-1}}\\ =\sqrt{\frac{1}{78-1}\left[{\left(111-108.9231\right)}^2+{\left(107-108.9231\right)}^2+\cdots +{\left(106-108.9231\right)}^2\right]}\\ =13.17097\end{array}$

Interpretation: The mean which is standardized for a large population of IQ is 100 and the mean calculated for provided number of observations is approximately 108.9231. The standard deviation for a large population is standardized to 15 and the calculated standard deviation for provided number of observationsis approximately 13.17097. From the calculation above, the mean of IQ scores among students is 9 points more than that of the overall population and the standard deviation of IQ scores among students is 2 points less than that of the overall population.

(b)

To determine

To find: The median of IQ scores and whether it is close to the mean or not.

Answer to Problem 82E

Solution: The median of IQ scores is 110. Yes, it is close to the mean.

Explanation of Solution

Calculation: The median can be calculated by following the steps below:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observation is 78, which is an even number.

Step 3: Median for an even number of observations is the average of ${\left(\frac{n}{2}\right)}^{th}$ and ${\left(\frac{n}{2}+1\right)}^{th}$ observations.

Step 4: The ${\left(\frac{n}{2}\right)}^{th}$ observation is the 39^th observation, which is 110, and the ${\left(\frac{n}{2}+1\right)}^{th}$ is the 40^th observation, which is110.

Step 5: The average of both of the observations is also 110, which is the median.

Interpretation: The median is 110 and the mean calculated in part (a) is 108.9231. As expected, the median is close to the mean, so the curve of probability distribution is more or less symmetric in this case.

(c)

To determine

To find: Mean and median for GPA data.

Answer to Problem 82E

Solution: The mean for the GPA data is 7.447 and the median is 7.829.

Explanation of Solution

Calculation: Mean for the GPA can be calculated as:

$\overline{x}=\frac{{\displaystyle \sum _{i=1}^n{x}_i}}{N}$

$\begin{array}{c}\overline{x}=\frac{7.94+8.292+4.643+\cdots +6.938}{78}\\ =\frac{580.83}{78}\\ =7.4465\end{array}$

The median can be calculated using the following steps:

Step 1: Arrange all the observations in ascending order.

Step 2: The number of observations is 78, which is an even number.

Step 3: Median for an even number of observations is the average of ${\left(\frac{n}{2}\right)}^{th}$ and ${\left(\frac{n}{2}+1\right)}^{th}$ observations.

Step 4: The ${\left(\frac{n}{2}\right)}^{th}$ observation is the 39^th observation, which is 7.825, and the ${\left(\frac{n}{2}+1\right)}^{th}$ is the 40^th observation which is 7.833.

Step 5: The average of both of the observations is 7.829, which is the median.

Interpretation: According to the stem plot in Exercise 1.39, the mean and median of GPA differ a bit from each other. More precisely, they are close to each and the distribution is skewed to the left.