91Ó°ÊÓ

Given in the question that,

Here are the weights (in milligrams) of $58$diamonds from a nodule carried up to the earth’s surface in surrounding rock. These data represent a single population of diamonds formed in a single event deep in the earth.

Consider the below table, for weights (in milligrams) of $58$diamonds.

A histogram shows that the data are skewed right, not symmetric. Hence, the distribution of weights of these diamonds is skewed to right. The median would then be a better measure of the center.

It is clear in the histogram that class $18-20$is an outlier because it falls outside the overall pattern. The spread of a data set refers roughly to how wide be the data relative to its center. The range is the difference between the maximum value and the minimum value in the data, or simply those two numbers. The spread is the range of values from the lowest value to the largest value. The spread is a more precise measure than the center.

Range= Highest value- Lowest value

$=33.8-0.1$

$=33.7$

$N=58$

$\frac{N}{2}=\frac{58}{2}$

$=29$

Cumulative frequency just greater than $29$, is$31$and the class corresponding to $31$is$4-6$is median class. Median is obtained by the formula:

$\text{Median}=l+\frac{h}{f}\left(\frac{N}{2}-c\right)$

Where:

$I$is the lower limit of the median,

$f$is the frequency of the median class

$\mathrm{h}$is the magnitude of the median class

$c$is the c.f. of the class preceding the median class

and $N=∑f$

Median

$=5.6$

Consider the following table:

$=\frac{452}{58}=7.79$

Skewness $\left({\mathrm{S}}_{\mathrm{k}}\right)=$Mean-Median

$=7.79-5.6$

$=2.19$

The first quartile is the median of the data values below the median. Since there are $29$data values below the median, the first quartile is the${15}^{\text{th}}$data value $Q_1=3.5$

The third quartile is the median of the data values above the median. Since there are $29$data values above the median, the first quartile is the ${15}^{\text{th}}$data value above the median, which corresponds with the $29+15={44}^{\text{th}}$data value in the sorted data set $Q_3=9$

The interquartile range $IQR$is the difference of the third and first quartile.

$$\begin{gathered} IQR=Q_3-Q_1 \\ =9-3.5 \\ =5.5 \end{gathered}$$

Shape: right skewed, because the highest bar in the histogram is to the left and the tall to the right.

Outliers: no, because there are no gaps in the histogram between bars.

Center: median is$5.6\mathrm{mg}$

Spread: interquartile range is$5.5\mathrm{mg}$