# Defining-mean,-median-and-mode

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```					Defining mean, median and mode and standard deviation When people talk about 'averages', they're usually referring to the mean. To calculate the mean, also known as the arithmetic mean, you:
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Add all the values together Divide the total by the number of values

The median is the middle or central value. If all the figures are put in order from lowest to highest, or vice versa, the median is the value that is positioned in the middle of all the numbers.

The mode is the value or figure that occurs most frequently. In this case 42 appears twice, while all the other values only appear once.

Mean, median and mode are all ways of displaying average or statistical values.

The arithmetic mean is the most common and best understood by the general public.

The median is the middle value in a set and the mode is the most frequently occurring value.

Images, and text abstracts taken from NLN Material - Mean, median, mode and standard deviation

Standard deviation shows how spread out the data is from the mean. The mean

Here's how you would calculate the mean of a long set of data.

First you add up all the values. Here, the total comes to 926. Then you divide the total by the number of values. There are 14 values, so 926 divided by 14 equals 66.14. So 66.14 is the mean.

Note that the mean is not always a whole number and may not be one of the values in the original data.

58+59+60+62+64+65+67+67+68+70+71+71+71+73= 926 926 = 66.14 14

Advantages - the mean is:
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The most commonly used and the most readily understood measure The favoured method for displaying data for the general public Easy to calculate and requires no preparation - like first putting values in order - before you can calculate it

Disadvantages - the mean:
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Can be misleading if there are a few extreme values Can't be used to summarise values that have already been summarised

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Can't be used where actual values are unknown, and data is presented in groups

Images, and text abstracts taken from NLN Material - Mean, median, mode and standard deviation

The median The median is the middle, or central value, once the data has been arranged in ascending order. Finding the median is straightforward if there is an odd number of values. In the first data set shown here, there are 15 values so the median is the eighth value along 67. With an even number of values the median is calculated as the average of the middle two values. In the second data set, the median falls between the eighth and ninth values, so 65 and 67 are added together and divided by 2, giving a median of 66. 58 59 60 62 64 64 65 67 67 68 70 71 71 71 73 57 58 59 60 62 64 64 65 67 67 68 70 71 71 71 73

The median is easy to calculate and relies only on the relative position of the values to each other. This means that it is not affected by extreme values.

Since the median does not rely on actual values, it can still be used when extreme data is recorded in groups - as in 'under 20' or '60 plus'.

The median does not take account of exact values. So the middle bar in a bar chart can be the one with the lowest number of counts and the median.

The median can be misleading when representing a small data set with a wide range of values. The central value may not reflect the spread of the data at all.

Images, and text abstracts taken from NLN Material - Mean, median, mode and standard deviation

The mode The mode is the most frequently occurring value in a data set. In the list of values shown below, the most frequently occurring value is 71. So in this case, the mode is 71.

But what about this data set? Here there are two most frequently occurring values - 64 and 71. When this happens, both are the mode and the data set is bimodal.

If there were more than two modes, it would be multimodal. The mode won't give you a central view of the data as the mean or median does. Sometimes, it will be a value from one extreme of the data range, but it can be very useful. For example, it will show what time of day a surgery is at its busiest, and therefore when it needs the most staff on-duty.

Images, and text abstracts taken from NLN Material - Mean, median, mode and standard deviation

Displaying the mode The mode, or modes, can be displayed visually on a graph or chart.

This bar chart shows the number of days patients have spent in hospital. The tallest bar is for three days, so this is the mode.

Here, the same number of patients stayed for three days as those who stayed for five days.

This graph is bimodal, because there are two most frequently occurring values.

This chart is multimodal because there are three most frequently occurring values.

This is because the same number of patients stayed for three days, five days and seven days.

Images, and text abstracts taken from NLN Material - Mean, median, mode and standard deviation

Standard deviation

These graphs show the waiting times for patients at different Accident & Emergency departments. Both have the same mean waiting time of 120 minutes, but the graph for Department A has a very steep curve while Department B has a more shallow curve. If you had to choose between these A&E departments, which would you choose? Would your chances of being seen after waiting for 120 minutes be better at Department A or Department B?

It is in answering questions like this that standard deviation is useful.

Images, and text abstracts taken from NLN Material - Mean, median, mode and standard deviation

Making sense of standard deviation Standard deviation, calculated using a complicated formula, is a measure of how tightly data is clustered around the mean. In most cases, standard deviation is calculated so that:
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One standard deviation is 68% of the sample Two standard deviations are 95% of the sample Three standard deviations are 99% of the sample

As you can see, the steeper the normal distribution curve on a graph, the smaller the standard deviation. In comparison, the shallow curve has a large standard deviation.

A small standard deviation shows that the data is clustered tightly around the mean.

Images, and text abstracts taken from NLN Material - Mean, median, mode and standard deviation

Revision Questions 1) Which of the following describes the term ‘bimodal’?
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The single most frequently occurring value The two most frequently occurring values The three or more most frequently occurring values The mean score is the same as the mode

2) What do you understand by the term 'mean'?
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The most frequently occurring value The middle value The average value The smallest value

3) Which of the following is a disadvantage of using the mean?
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Extreme values can distort it It is difficult to calculate It is not generally understood It is easily confused with the average

4) Which of the following is an advantage of using the median?
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The public understand it better than the mean or mode It is good at representing extreme data accurately You don’t need exact values to calculate it You don’t need to do any preparation before calculating it

5) What is the approximate percentage of the values that are between the mean score and one standard deviation above the mean?
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34% 68% 95% 99%

6) Here is the standard deviation (expressed in number of rings) from a mean response time of 3, of four telephone banking services. Which is the most reliable service?
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Service A - 6 rings Service B - 3 rings Service C - 7 rings Service D - 2 rings

Images, and text abstracts taken from NLN Material - Mean, median, mode and standard deviation

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