Mean Deviation & Standard Deviation

Today I will be writing about the differences between a mean deviation and a standard deviation.

Mean Deviation

Say that x is a set of data and that this set of data contains the values 1, 2, 3, 4 and 5.  Now, we’d both be able to agree that this set of data contains 5 values; furthermore, we’d be able to calculate the mean of this set of data by performing the calculation below:

\overline { x } =\frac { \Sigma x }{ n } =\frac { 1+2+3+4+5 }{ 5 } =\frac { 15 }{ 5 } =3

Fairly straightforward, right? Ok, but how would we go about calculating the mean deviation of this set of data? Well, firstly we’d have to figure out the distances between each individual value in the set of data and the mean of the set of data. Secondly, we’d have to sum up all the distances between the values in the set of data and the mean of the set of data. Lastly, we’d then have to divide this total sum by the amount of values contained within the set of data – and in this case it would be 5.

The mean deviation calculation would look like this:

\frac { \Sigma \left| x-\overline { x } \right| }{ n } \\ \\ =\frac { \left| 1-3 \right| +\left| 2-3 \right| +\left| 3-3 \right| +\left| 4-3 \right| +\left| 5-3 \right| }{ 5 } \\ \\ =\frac { 2+1+0+1+2 }{ 5 } \\ \\ =\frac { 6 }{ 5 } \\ \\ =1.2

So, what the mean deviation of a set of data tells us is how far away values in the set of data lie away from the mean – on average. In this case, values in the set of data lie a distance of 1.2 away from the mean of the set of data – on average.

We can actually create a diagram which would neatly demonstrate what a mean deviation would like. The diagram below relates to the set of data I’ve been writing about…

mean deviation

Standard Deviation

Now, to find the standard deviation of a set of data we must perform an entirely different calculation. A standard deviation is certainly not the same as a mean deviation.

To demonstrate what a standard deviation is, I’m going to continue working with the set of data I used to describe what a mean deviation is.

To get the standard deviation of the set of data we were working with, we must first subtract the mean of the set of data from each value in our set of data. Secondly, we must square up each of our results. Once we’ve done this, we have to sum up the values of the results which have been squared. Thirdly, we must divide this total sum by the amount of values which exist in our set of data – and we already know that this value is 5. What we have at the moment is the variance of our set of data. It turns out that the standard deviation of a set of data is in fact the square root of the variance of the set of data

This means that our standard deviation calculation would look like this…

\sigma =\sqrt { \frac { \Sigma { \left( x-\overline { x } \right) }^{ 2 } }{ n } } \\ \\ =\sqrt { Variance } \\ \\ =\sqrt { \frac { { \left( 1-3 \right) }^{ 2 }+{ \left( 2-3 \right) }^{ 2 }+{ \left( 3-3 \right) }^{ 2 }+{ \left( 4-3 \right) }^{ 2 }+{ \left( 5-3 \right) }^{ 2 } }{ 5 } } \\ \\ =\sqrt { \frac { { \left( -2 \right) }^{ 2 }+{ \left( -1 \right) }^{ 2 }+{ \left( 0 \right) }^{ 2 }+{ \left( 1 \right) }^{ 2 }+{ \left( 2 \right) }^{ 2 } }{ 5 } } \\ \\ =\sqrt { \frac { 4+1+0+1+4 }{ 5 } } \\ \\ =\sqrt { \frac { 10 }{ 5 } } \\ \\ =\sqrt { 2 }

The square root of 2 is roughly 1.414.

Differences between a Mean Deviation and a Standard Deviation

So, what we have found out is that the mean deviation of a set of data isn’t really related to the standard deviation of a set of data. The same set of data produced two different results… We obtained the value 1.2 as a mean deviation and 1.414 as a standard deviation in our set of data. The two values we’ve obtained are relatively close to one another but aren’t the same, however, standard deviation calculations are incredibly important and are used by economists, engineers and scientists.

Finally, we discovered two other interesting facts…

(1) Variance is equal the distance from the mean in a set of data to individual values in a set of data squared, summed up and divided by the amount of values which exist in this particular set of data.

(2) The standard deviation of a set of data is equal to the square root of variance in a set of data. This is because a standard deviation must be a scalar quantity, in other words a length. Variances are produced with distances between values in a set of data and their means squared, which in fact produce two dimensional objects (squares), and this is precisely why we have to get the square root of variance in order to figure out what the standard deviation in a set of data is. Variances are related to two dimensional objects, whilst standard deviations are related to one dimensional objects i.e lengths or distances.

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