Question

Diagram 15 shows the number of books read by 18 students in a library in a certain month

Answer 1

`\begin{gathered} a)23 \\ b)13 \\ c)54.11 \\ d)7.36 \end{gathered}`

Step-by-step explanation

First, let us sort the data according to size:

`\begin{gathered} 6,7,8,8,9,10,12,13,14 \\ 16,16,16,17,22,25,27,28,29 \end{gathered}`

a) To find the range, find the difference between the largest and least data point from the data set.

Therefore, the range is:

`\begin{gathered} R=29-6 \\ R=23 \end{gathered}`

b) To find the interquartile range, first, find the first quartile and the third quartile. Then find the difference between them.

Since the data set has an even number of data points, the first quartile is the middle number in the first half of the data set.

The first half of the data set ranges from 6 to 14. The middle number in that range is 9.

The third quartile is the middle number in the second half of the data set. The second half of the data set ranges from 16 to 29. The middle number in that range is 22.

Hence, the first quartile is 9 and the third quartile is 22.

Therefore, the interquartile range is:

`\begin{gathered} IQR=22-9 \\ IQR=13 \end{gathered}`

c) To find the variance, we have to apply the formula:

`\sigma^2=(\sum (x-\mu)^2)/(N)`

where μ = mean of data set.

The mean of the data set is:

`\begin{gathered} \mu=(6+7+8+8+9+10+12+13+14+16+16+16+17+22+25+27+28+29)/(18) \\ \mu=15.72 \end{gathered}`

Now, we have to find the difference between each term and the mean, find the squares, and then sum them.

Let us put it in a table:

Now, to find the variance, divide that sum by the number of values in the data set. That is:

`\begin{gathered} \sigma^2=(\sum (x-\mu)^2)/(N) \\ \sigma^2=(974)/(18) \\ \sigma^2=54.11 \end{gathered}`

That is the variance.

d) To find the standard deviation, find the square root of the variance:

`\begin{gathered} \sigma=\sqrt[]{\sigma^2} \\ \sigma=\sqrt[]{54.11} \\ \sigma=7.36 \end{gathered}`

That is the standard deviation.