Question

ShakespeareThe histogram shows the distribution of lengths of words used in Shakespeare’s plays.

(a) Find the median word length in Shakespeare’s plays from the histogram. Explain your method clearly.

(b) Calculate the mean of the distribution. Show your work.

Short answer

Part (a) 4 words.

Part (b) 4.13 words

Step-by-step solution

Part (a) Step 1. Given information.

The given histogram is shown below with their frequency:

Part (a) Step 2. Find the median word length in Shakespeare’s plays from the histogram. 

As we know that median is the middle value of a data set. As the given set is even, the median will be the average of two middle values (50th and 51st) of this sorted data.

Because $5+17+22=44$(sum frequencies first three bars) and $5+17+22+23=67$(sum frequencies first four bars), the 50th and 51st data values are both equal to 4 in the histogram's third bar.

$$\begin{gathered} M=\frac{4+4}{2} \\ =4 \end{gathered}$$

The median units are the same as the data values' units, so the median is 4 words.

Part (b) Step 1. Calculate the mean of the distribution.

Word Length (x)	Frequency (f)	fx
1	5	5
2	17	34
3	22	66
4	23	92
5	12	60
6	7	42
7	5	35
8	4	32
9	3	27
10	2	20
Total	100	413

$$\begin{gathered} \overline{x}=\frac{\sum \left(fx\right)}{n} \\ =\frac{413}{100} \\ =4.13 \end{gathered}$$