Chapter 2, Problem 1TP

To determine

The mean, median and mode for the given data.

Answer to Problem 1TP

The mean of the given data is 28.375, the median is 25.5 and the mode is 23.

Explanation of Solution

The given information represents the set of 40 reading scores.

31	32	43	42
24	34	25	44
23	43	24	36
25	41	23	28
14	21	24	17
25	23	44	21
13	26	23	32
12	26	14	42
14	31	52	12
23	42	32	34

Calculation:

The formula for mean is given below:

$\text{Mean}=\frac{{\displaystyle \sum _1^n{X}_n}}{n}$

$\begin{array}{c}{\displaystyle \sum _1^{40}{X}_n}=31+24+23+\mathrm{...}+42+12+34\\ =1,135\end{array}$

Substitute 1,135 for ${\displaystyle \sum _1^{40}{X}_n}$ and 40 for n in the above equation,

$\begin{array}{c}\text{Mean}=\frac{1,135}{40}\\ =28.375\end{array}$

Thus, the mean of the given data is 28.375.

Arrange the given data in ascending order, to find the median of the given data.

S. No	Values	S. No	Values	S. No	Values	S. No	Values
1	12	11	23	21	26	31	36
2	12	12	23	22	26	32	41
3	13	13	23	23	28	33	42
4	14	14	23	24	31	34	42
5	14	15	24	25	31	35	42
6	14	16	24	26	32	36	43
7	17	17	24	27	32	37	43
8	21	18	25	28	32	38	44
9	21	19	25	29	34	39	44
10	23	20	25	30	34	40	52

The total number of terms is 40, which is an even number.

The formula for median when n is even is given below:

$\text{Median}=\frac{\frac{n}{2}\text{th}\text{term}+\left(\frac{n}{2}+1\right)\text{th}\text{term}}{2}$

Substitute 40 for n in the above equation,

$\begin{array}{c}\text{Median}=\frac{{\frac{40}{2}}^{\text{th}}\text{term}+{\left(\frac{40}{2}+1\right)}^{\text{th}}\text{term}}{2}\\ =\frac{{\text{20}}^{\text{th}}\text{term}+{\left(20+1\right)}^{\text{th}}\text{term}}{2}\\ =\frac{{\text{20}}^{\text{th}}\text{term}+{21}^{\text{st}}\text{term}}{2}\end{array}$

The 20^th term is 25 and 21^st term is 26 when the data is arranged in the ascending order.

Substitute 25 for $\text{20th}\text{term}$ and 26 for $21\text{st}\text{term}$ in the above equation,

$\begin{array}{c}\text{Median}=\frac{25+26}{2}\\ =\frac{51}{2}\\ =25.5\end{array}$

Thus, the median of the given data is 25.5.

Mode:

The value in data set which occurs maximum times is called mode.

The frequency of the data is tabulated below:

Value	Frequency	Value	Frequency
12	2	31	2
13	1	32	3
14	3	34	2
17	1	36	1
21	2	41	1
23	5	42	3
24	3	43	2
25	3	44	2
26	2	52	1
28	1	-	-

From the above table it is noticed that the highest frequency is 5 for 23.

Thus, the mode of the given data is 23.