Q. 3.174 - Elementary Statistics

Question

Medieval Cremation Burials. In the article "Material Culture as Memory: Combs and Cremations in Early Medieval Britain" (Early Medieval Europe, Vol. $12$. Issue $2$. pp. $89-128$), H. Williams discussed the frequency of cremation burials found in $17$ archaeological sites in eastern England. Here are the data.

a. obtain and interpret the quartiles.

b. determine and interpret the interquartile range.

c. find and interpret the five number summary.

d. identify potential outliers, if any

e. construct and interpret a boxplot.

Step-by-Step Solution

Answer

(a) The three quartiles are $Q_1=46,Q_2=83,Q_3=265$.

(b) The interquartile range is $219$.

(c) The variation in first quarter is less but in last quarter is high.

(d) The potential outlier is $2484$.

(e) See the boxplot in step $10.$

Part (a) Step 1: Given Information

We are given that H. Williams discussed the frequency of cremation burials found in $17$ archaeological sites in eastern England. Here are the data and we have to find out all the quartiles.

Part (a) Step 2: Explanation

First, organize the frequencies in ascending order and after that we get $21,34,35,46,46,48,51,64,83,86,119,258,265,385,429,523,2484$ then find the median of entire data as no. of observations is 21, so the median will be at $\frac{17+1}{2}=9th$ so we get $83$ as the median which is the $Q_2$.

Divide it in half and include median in both as observations are odd.

We get, Bottom Half is $21,34,35,46,46,48,51,64,83$

Top Half is $83,86,119,258,265,385,429,523,2484$                                                                        Then find the median of bottom half and median is at the position $\frac{9+1}{2}=5$ and that means median is at $5th$ position which is $46$ and this is $Q_1$  and last find the median of the top half and median is at position $5th$ and that means median is $265$ and that is $Q_3$.

Hence, these are the all quartiles.

Part (b) Step 1: Given Information

We are given that H. Williams discussed the frequency of cremation burials found in $17$ archaeological sites in eastern England. Here are the data and we have to find out the interquartile range.

Part (b) Step 2: Explanation

The interquartile range is the difference between the first and the third quartiles,

By which we get, IQR$=Q_3-Q_1$

Putting the values, 

We get,$IQR=265-46=219$

Hence this is the required interquartile range.

Part (c) Step 1: Given Information

We are given that H. Williams discussed the frequency of cremation burials found in $17$  archaeological sites in eastern England. Here are the data and we have to find out the five number summary and also interpret it.

Part (b) Step 2: Explanation

The five-number summary of a data set is Min ,$Q_1,Q_2,Q_3$, Max

We get, Min$=21,Q_1=46,Q_2=83,Q_3=265$, Max$=2484$

Now to get the variation subtract lowest from highest  by which we get,$25,37,182,2219$ and according to it first quarter has less variation but last quarter has high variation.

Part (d) Step 1: Given Information

We are given that H. Williams discussed the frequency of cremation burials found in $17$ archaeological sites in eastern England. Here are the data and we have to find out the potential outliers.

Part (d) Step 2: Explanation

First find the lower and upper limit which is,

Lower limit $=Q_1-1.5\times IQR$ and Upper Limit$=Q_3+1.5\times IQR$

Put the values of all variable we get,

lower limit$=46-1.5\times 219=-279.5$ and upper limit $=265+1.5\times 219=593.5$.

And potential outlier is defined as a value that is  outside of lower and upper limit and according to this question only $2484$ is outside the upper limit. So, this is the potential outlier.

Part (e) Step 1: Given Information

We are given that H. Williams discussed the frequency of cremation burials found in $17$ archaeological sites in eastern England. Here are the data and we have to interpret the boxplot.

Part (e) Step 2: Explanation

First to make the boxplot we need quartile, potential outliers and adjacent values.

and from previous parts we have, $Q_1=46,Q_2=83,Q_3=265$ and potential outlier is $2484$ and adjacent values are $34$ and $523$ now plot these

Hence this is the required boxplot which means that first and last lines are adjacent values and mid box is representing quartiles.