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Measures of Central Tendency with some MS Excel Tips

The Means, Medians and Modes of Grouped Data

Introduction

Let's look briefly at the measures of central tendency associated with grouped data. What does that mean? Well, sometimes we find data that is presented in grouped form, not as individual data points. The following table contains grouped data:

Classes Frequency
0 5 6
5 10 12
10 15 19
15 20 3
Total 40

Let's assume that the classes refer to the ages of our sample: 0 5 means that we have gathered data here for anyone who is older than 0 but younger than or exactly equal to 5 years old. 5 10 means that we have looked at someone who is in the class or group of people who are older than 5 years but younger than or exactly equal to 10 years and so on.

In the frequency column, we find that 6 people are between 0 and 5 years of age, 14 people are between 5 and 10 years of age and altogether, we have data on 40 people spanning the entire age range 0 to 20 years.

Arithmetic Mean of Grouped Data

To calculate the arithmetic mean of grouped data, we need to extend the above table as follows:

Classes Frequency Mid Point
of Class
Age (years) f X fX
0 5 6 2.5 15
5 10 12 7.5 90
10 15 19 12.5 237.5
15 20 3 17.5 52.5
Total 40 395

The arithmetic mean is

The Median of Grouped Data

The median is the middle value of a data set and for grouped data, we can find the class that the median resides in relatively easily. In the case of the example we used for the arithmetic mean of grouped data, we can see that the median value is the average of the 20th and 21st values there are 40 data points, an even number of data points. The median class is highlighted in the following table:

Classes Frequency
Age (years) f
0 5 6
5 10 12
10 15 19
15 20 3
Total 40

The calculation of the median of grouped data is based on the following formula

Looks a bit hideous don't you think? Let's look at it in detail.

L = the lower limit of the class containing the median
n = the total number of frequencies
f = the frequency of the median class
CF = the cumulative number of frequencies in the classes preceding the class containing the median
i = the width of the class containing the median

Putting the numbers from the example into the formula now, we see that the median value is 10.53:

The Mode of Grouped Data

The mode is, very simply, the mid point f the class containing the largest number of class frequencies.

Using the previous example again, we find the mode there is:

The class containing the largest number of class frequencies is highlighted below The mid point of the modal class is 12.5 so the mode of these data is 12.5

Classes Frequency
Age (years) f
0 5 6
5 10 12
10 15 19
15 20 3
Total 40

In summary

  • the arithmetic mean of the grouped data is 9.875 years
  • the median of the grouped data is 10.53 years
  • the mode of the grouped data is 12.5 years

    As a matter of interest, if we were now to find that the full, ungrouped, data is as follows, we can see how accurate the results we have just found are:

    2 6 14 13
    4 9 14 15
    2 10 12 11
    5 6 14 15
    0 6 14 14
    3 10 15 15
    9 9 13 11
    6 8 15 17
    10 14 14 16
    10 11 13 18

  • Arithmetic mean = 11
  • Median = 11
  • Mode = 14

    Conclusion

    That is the end of the four part series on averages. We have tken a relatively detailed look at the arithmetic weighted and geometric means together with the median and the mode. We have also investigated the averages concerned with group data.

    Overall, this section has provided us with a mixture of straightforward statistics and an introduction to some of the issues with which Microsoft Excel can help us.

    Other pages in this section:

    Hop back to the Arithmetic Mean page

    Slither in to the Median and Mode page

    Huff 'n Puff your way over to the Geometric Mean page

    Duncan Williamson
    24 February 2002

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    Webmaster Duncan Williamson 2002