Calculate the mean, median and mode of the following numbers:

(i) 17, 19, 11, 23, 19

(ii) 7, 9, 8, 11, 8, 12, 8, 9

(iii) 2, 1, 0, 3, 1, 2, 3, 4, 3, 5

(iv) 8, 10, 7, 6, 10, 11, 6, 13, 10

(i) Arranging given observations in ascending order:

11, 17, 19, 19, 23

Sum of observations = 17 + 19 + 11 + 23 + 19 = 89.

By formula,

Mean = $\dfrac{\text{Sum of observation}}{\text{No. of observation}} = \dfrac{89}{5}$ = 17.8

Here n = 5, which is odd.

By formula,

Median = $\dfrac{\text{n} + 1}{2} \text{ th observation}$

$= \dfrac{5 + 1}{2} \text{ th observation} \\[1em] = \dfrac{6}{2} \text{ th observation} \\[1em] = 3 \text{ rd observation} \\[1em] = 19.$

From set of observations, we see that:

19 has the maximum frequency.

Hence, mean = 17.8, median = 19, mode = 19.

(ii) Arranging given observations in ascending order:

7, 8, 8, 8, 9, 9, 11, 12

Sum of observations = 7 + 8 + 8 + 8 + 9 + 9 + 11 + 12 = 72

By formula,

Mean = $\dfrac{\text{Sum of observation}}{\text{No. of observation}} = \dfrac{72}{8}$ = 9

Here n = 8, which is even.

By formula,

Median = $\dfrac{\dfrac{\text{n}}{2} \text{ th observation} + \Big(\dfrac{\text{n}}{2} + 1\Big) \text{ th observation}}{2}$

$= \dfrac{\dfrac{8}{2} \text{ th observation} + \Big(\dfrac{8}{2} + 1\Big) \text{ th observation}}{2} \\[1em] = \dfrac{4 \text{ th observation} + (4 + 1) \text{ th observation}}{2} \\[1em] = \dfrac{4 \text{ th observation} + 5 \text{ th observation}}{2} \\[1em] = \dfrac{8 + 9}{2} \\[1em] = \dfrac{17}{2} \\[1em] = 8.5.$

From set of observations, we see that :

8 has the maximum frequency.

Hence, mean = 9, median = 8.5, mode = 8.

(iii) Arranging given observations in ascending order:

0, 1, 1, 2, 2, 3, 3, 3, 4, 5

Sum of observations = 0 + 1 + 1 + 2 + 2 + 3 + 3 + 3 + 4 + 5 = 24

By formula,

Mean = $\dfrac{\text{Sum of observation}}{\text{No. of observation}} = \dfrac{24}{10}$ = 2.4

Here n = 10, which is even.

By formula,

Median = $\dfrac{\dfrac{\text{n}}{2} \text{ th observation} + \Big(\dfrac{\text{n}}{2} + 1\Big) \text{ th observation}}{2}$

$= \dfrac{\dfrac{10}{2} \text{ th observation} + \Big(\dfrac{10}{2} + 1\Big) \text{ th observation}}{2} \\[1em] = \dfrac{5 \text{ th observation} + (5 + 1) \text{ th observation}}{2} \\[1em] = \dfrac{5 \text{ th observation} + 6 \text{ th observation}}{2} \\[1em] = \dfrac{2 + 3}{2} \\[1em] = \dfrac{5}{2} \\[1em] = 2.5.$

From set of observations, we see that:

3 has the maximum frequency.

Hence, mean = 2.4, median = 2.5, mode = 3.

(iv) Arranging given observations in ascending order:

6, 6, 7, 8, 10, 10, 10, 11, 13

Sum of observations = 6 + 6 + 7 + 8 + 10 + 10 + 10 + 11 + 13 = 81

By formula,

Mean = $\dfrac{\text{Sum of observation}}{\text{No. of observation}} = \dfrac{81}{9}$ = 9

Here n = 9, which is odd.

By formula,

Median = $\dfrac{\text{n} + 1}{2} \text{ th observation}$

$= \dfrac{9 + 1}{2} \text{ th observation} \\[1em] = \dfrac{10}{2} \text{ th observation} \\[1em] = 5 \text{ th observation} \\[1em] = 10.$

From set of observations, we see that:

10 has the maximum frequency.

Hence, mean = 9, median = 10, mode = 10.