(MEDIAN, MEAN, MODE) FOR MBA 1st SEM

Measures of average in grouped and continuous data

The mean

We already know how to find the mean from a frequency table. Finding the mean for grouped or continuous data is very similar.
The grouped frequency table shows the number of CDs bought by a class of children in the past year.

 

Number of CDs	Frequency (f)
0-4	10
5-9	12
10-14	6
15-19	2
>19	0

• We know that 10 children have bought either 0, 1, 2, 3 or 4 CDs, but we do not know exactly how many each child bought.
• If we assumed that each child bought 4 CDs, it is likely that our estimate of the mean would be too big.
• If we assumed that each child bought 0 CDs, it is likely that our estimate would be too small.
• It therefore seems sensible to use the mid-point of the group and assume that each child bought 2.

Finding the mid-points of the other groups, we get:

 

Number of CDs	f	Mid-point, x	fx
0-4	10	2	20
5-9	12	7	84
10-14	6	12	72
15-19	2	17	34
>19	0	-	0

The mean is
Remember: This is only an estimate of the mean.

The median

As explained previously, the median is the middle value when the values are arranged in order of size.
As the data has been grouped, we cannot find an exact value for the median, but we can find the class which contains the median.

 

Number of CDs	Frequency (f)
0-4	10
5-9	12
10-14	6
15-19	2
>19	0

There are 30 children, so we are looking for the class which contains the (30 + 1) ÷ 2 = 15¹₂th value. The median is therefore within the 5-9 class.

The mode

The mode is the most common value.
We cannot find an exact value for the mode, and therefore give the modal class. The modal class is 5-9.

Advantages and disadvantages of mean, median and mode

With three averages to choose from mean, median and mode – which should we use?
The following table shows the advantages and disadvantages of these different averages.

 

Average	Advantages	Disadvantages
Mean	All the data is used to find the answer	Very large or very small numbers can distort the answer
Median	Very big and very small values don't affect it	Takes a long time to calculate for a very large set of data
Mode or modal class	The only average we can use when the data is not numerical	There may be more than one mode There may be no mode at all if none of the data is the same It may not accurately represent the data

Example
This table shows the annual salary of people who work at a garden centre.

 

Annual salary (£)	Number of people
0 - 9,999	10
10,000 - 19,999	9
20,000 - 29,999	9
30,000 - 39,999	1
40,000 - 49,999	1

The modal class is £0 - £9,999.
Question
What is the disadvantage of using the modal class?
Answer
Even though the range £0 - £9,999 contains the most number of people, the next two ranges have comparable numbers and so is not representative of the data.
Question
What is the disadvantage of using the mean?
Almost everyone earns under £30,000. The mean would be distorted by the fact that two people earn much more than this.