Question

Find the median and mode:

xi	1	2	3	4	5	6	7	8	9	10
fi	1	2	3	3	6	10	5	4	3	3

Answer 1

To find the median and mode for the data provided in the table, we first need to organize the values and calculate the cumulative frequency.

Frequency distribution table

Value (xi)	Frequency (fi)	Cumulative Frequency (cf)	fi × xi
1	1	1	1
2	2	3	4
3	3	6	9
4	3	9	12
5	6	15	30
6	10	25	60
7	5	30	35
8	4	34	32
9	3	37	27
10	3	40	30
Total	N = 40		Σfixi = 240

1. Finding the mode

The mode is the value that appears most frequently in the data set.

Looking at the frequency (f_i) column, the highest frequency is 10.

This frequency corresponds to the value x_i = 6.

Mode = 6

2. Finding the median

Since the total frequency N is 40 an even number, the median is the average of the `(N/2)^(th)` and `(N/2 + 1)^(th)` observations.

1. Calculate positions: `40/2` = 20^th and 20 + 1 = 21^st positions.

2. Locate these in the cumulative frequency (cf):

1. The 15^th observation is 5.
2. Observations from the 16^th to the 25^th are all 6.

3. Both the 20^th and 21^st observations are 6.

Median = 6

3. Finding the mean

The second part of your image asks for the mean.

We can calculate this using the formula `barx = (sumf_ix_i)/N`:

`barx = 240/40`

`barx = 6`

Mean = 6