Question

Find the mean, median and mode of the following data: 

Class	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50	50 – 60	60 – 70
Frequency	6	8	10	15	5	4	2

Answer 1

1. Construct the calculation table

To find the statistical measures, we first need to determine the class mark (midpoint x_i), the product of frequency and midpoint (f_ix_i) and the cumulative frequency (cf) for each class interval.

Class Interval	Frequency (fi)	Midpoint `bb(((x_i = "Lower" + "Upper")/2))`	fixi	Cumulative Frequency (cf)
0 – 10	6	5	30	6
10 – 20	8	15	120	14
20 – 30	10	25	250	24
30 – 40	15	35	525	39
40 – 50	5	45	225	44
50 – 60	4	55	220	48
60 – 70	2	65	130	50
Total	Σfi = 50		Σfixi = 1500

2. Compute the arithmetic mean

The arithmetic mean for grouped data is calculated using the direct method formula:

`barx = (sumf_ix_i)/(sumf_i)`

Substitute the values from our table:

`barx = 1500/50 = 30`

3. Compute the median

First, find the median class.

Total frequency N = 50, so `N/2 = 50/2 = 25`.

Looking at the cumulative frequency (cf) column, the first class interval where the cumulative frequency exceeds or equals 25 is the 30 – 40 group.

Now, apply the median formula:

Median = `L + ((N/2 - cf_"prev")/f) xx h`

Where:

L (lower limit of the median class) = 30

`N/2 = 25`

`cf_"prev"` (cumulative frequency of the preceding class) = 24

f (frequency of the median class) = 15

h (class width) = 10

Substitute these values:

Median = `30 + ((25 - 24)/15) xx 10`

Median = `30 + (1/15) xx 10`

= 30 + 0.67

= 30.67

4. Compute the mode

First, identify the modal class, which is the class with the highest frequency. The highest frequency is 15, which corresponds to the 30 – 40 class interval.

Apply the mode formula:

Mode = `L + ((f_1 - f_0)/(2f_1 - f_0 - f_2)) xx h`

Where:

L (lower limit of the modal class) = 30

f₁ (frequency of the modal class) = 15

f₀ (frequency of the preceding class) = 10

f₂ (frequency of the succeeding class) = 5

h (class width) = 10

Substitute these values:

Mode = `30 + ((15 - 10)/(2(15) - 10 - 5)) xx 10`

Mode = `30 + (5/(30 - 15)) xx 10`

Mode = `30 + (5/15) xx 10`

= `30 + 10/3`

= 33.33