Answer to Question #150981 in Statistics and Probability for P

Question

Calculate the Median and Mode from the following data:
Annual Sales(Rs.000) Frequency
Less than 10 4
Less than 20 20
Less than 30 35
Less than 40 55
Less than 50 62
Less than 60 67

Answer 1

We have:

The median is the middle value, which in this case is the 34^th one, which is in the 20-29 group. So we can say "the median group is 20-29".

$$median =L+\frac{\frac{n}{2}-B}{G}\cdot w$$

where

L is the lower class boundary of the group containing the median

n is the total number of values

B is the cumulative frequency of the group before the median group

G is the frequency of the median group

w is the group width

$$median =19.5+\frac{\frac{67}{2}-20}{15}\cdot 10=28.5$$

The mode is the value that appears most often in a set of data values.

So we can say "the modal group is 30-39". But the actual mode may not even be in that group! Or there may be more than one mode. We don't really know without the raw data.

But, we can estimate the Mode using the following formula:

$$mode=L+\frac{f_m-f_{m-1}}{(f_m-f_{m-1})+(f_m-f_{m+1})}\cdot w$$

where

L is the lower class boundary of the modal group

f_m is the frequency of modal group

f_m-1 is the frequency of the group before the modal group

f_m+1 is the frequency of the group after the modal group

w is the group width

$$mode=29.5+\frac{20-15}{20-15+20-7}\cdot 10=32.2778$$