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Median and Mode

Go to the Geometric Mean page

Work through the Means, Medians and Modes of Grouped Data

Introduction

The most commonly used word for a measure of central tendency is the word average: this is the word we use when we want to know, actually to estimate, the number that is in the middle of a batch of data, or set of numbers.

A good example of an average is that we might want to know the average value of the houses for sale on a new housing development. If we find that there are six houses for sale at the moment and they have the values of 50,000, 50,000, 65,000, 67,500, 67,500 and 75,000, we would probably all say that the average selling price is 53,572, as shown in the table below:

House number	Selling price
1	50,000
2	50,000
3	65,000
4	67,500
5	67,500
6	75,000
average	53,572

We calculated this average by adding together all of the house prices and dividing them by the number of houses, six.

For every day use, this is the value for average we use and it is good enough. However, there are times when this version of average is not good enough.

There are 4 pages in this section and they discuss the arithmetic mean, the version of average we have just discussed, the weighted mean, the median, the mode and the geometric mean. We will also look briefly at the mean, median and mode for grouped data: where data comes in batches rather than one by one!

This section contains the definitions and descriptions of the Arithmetic Mean and the Weighted Mean. The Median and Mode are found on page 2 of this section. The Geometric Mean is on page 3 of this section; and the Mean, Median and Mode of Grouped Data are found on page 4 of the section.

Microsoft Excel: where appropriate, we will give you the MS Excel function that will give you the answer you are looking for too! (Please note, I am using Excel XP; and if you are using an earlier version be aware that you might not find all of the functions I discuss here.)

Arithmetic Mean

The word mean means average; and we can find the word mean being used in this context on maps. For example, if we look at an Ordinance Survey map of any town on a sea coast, we will find three values for the Tide: low tide, high tide and mean tide; with mean being the average. If we talk about the distance of the Moon from the earth, an astronomer will give us the mean distance: the average distance.

Definition: the arithmetic mean is a single value that summarises a set of data. It locates the centre of the values. The arithmetic mean is commonly shown as and the formula for calculating it is:

Example: Let's work out the arithmetic mean value of the number of overtime hours worked by the mechanics in a garage for the month of February 2002. The hours they worked were:

Grade 1	13	13	12	15	7	15	5	12
Grade 2	6	7	12	10	9	13	12

Calculate the arithmetic means as follows:

a) only for grade 1 mechanics b) only for grade 2 mechanics c) for all mechanics

Do you agree that

a) grade 1 mechanics worked an average of 11.5000 hours overtime and
b) grade 2 mechanics worked an average of 9.8571 hours overtime and
c) overall, the average number of overtime hours worked was 10.7333?

Microsoft Excel's mean function is =AVERAGE(range of data). Just enter your data in a column or a row and tell Excel where it is and it will use the AVERAGE function to determine the arithmetic mean for you.

So, if you had a column of numbers in the range A1 to A15, you would enter =AVERAGE(A1:A15) in the cell in which you want the average value to appear.

If you have your data in, say column A and column B, you could enter =AVERAGE(A1:B6), assuming your data are in the range A1 to B6.

Let's stay with the idea of the arithmetic mean and develop it a little into what is known as the weighted mean.

The Weighted Mean

In some cases when we want to calculate the arithmetic mean, we find that some observations take the same value. For example, I am 1.88 metres tall and I am not the only person who is that tall. So, when we take a data set of the heights of men, we might find:

Name	Height (metres)
Jack	1.88
John	1.75
Sam	1.88
Bill	1.63
George	1.75
Average	1.778

We can also represent the weighted mean as

In the case of our five men, is

Please note that if we had used the arithmetic mean method, we would have got the same result. Please also note that if we were dealing with a much larger data set and one in which, say, contained the heights of 50 men, 10 of whom were 1.88m, 25 of whom were 1.75m … we would save a lot of time by using the weighted mean method rather than the arithmetic mean method.

For you to do: evaluate the gate receipts from a football match, using the weighted mean method, given the following information.

15,000 people paid £15
16,000 people paid £20
3,000 people paid £30
1,000 people paid £50

as you work through this example, imagine using the arithmetic mean method here, with 35,000 values to consider!

Do you agree that the total gate receipts for this football match were 68,500?Well, they weren't! The answer is 685,000: I'm allowed a little jke from time to time aren't I?

Do you agree that the weighted mean is

There is no direct function in Microsoft Excel for the weighted mean. However, if you set up the range with w in column A and X in column B then wX in column C, you can use the AVERAGE function to calculate the weighted mean for you.

Problems with the Mean

The mean is a useful measure of central tendency for every day use. However, because it always uses every single value in the data series, it can run into problems when, for example, most of the numbers in a data set are fairly well related but a few of them are not.

What does this mean? Well, imagine that a group of bird watchers (ornithologists) spend their weekends logging the activities of birds in their own part of town. We then want to calculate the average number of birds spotted by the group; but the data we have are as follows:

Name	Birds
Jack	150
John	125
Sam	110
Bill	130
George	734
Average	249.8

Here we see that one value, we can call it an outlier, is significantly different to all of the other values and it has distorted our understanding of the word average or mean. Four of our bird watchers have observed an average of 128.75 birds; but then George came along with his huge score and significantly distorted the average.

In reality there is little that we can do about outliers with genuine data. After all, if George really did spot 734 birds, it must be true! However, if we are dealing with a large data set, we can often dismiss the outliers by ignoring them since their influence might not be so great in such a large data set.

Alternative measure of central tendency can help us to overcome the outlier problem, too: the median and the mode will help us here.

Now you have a choice:

Go to the Median and Mode page; or

Go to the Geometric Mean page

Work through the Means, Medians and Modes of Grouped Data

Duncan Williamson
24 February 2002