Mammoth Memory

Standard deviation

The graph below shows a thousand students taking a test in Class 3 and Class 4. You can see nearly 150 students got a score of between 80 and 90 marks in Class 3. In class 4 nearly 250 students got a score between 80 and 90.

Lets look at the mean (add up all the numbers and divide by the number of numbers) of the following graph.

Histogram of class 3 test scores

Histogram of class 4 test scoresThe mean of the test scores for year 3 is 85 and the mean of the test scores of year 4 are also 85.

The graphs have the same mean value but very different distributions.

One way to show the differences in these graphs non graphically is to give you a number that tells you how far away 68% of all the data is away from the mean (called 1 standard deviation).

1 Standard deviation of class 3 test scores shown on histogram

1 Standard deviation of class 4 test scores shown on histogram

This only works for graphs that have the shape of a bell, or a bell curve.

Graphs with a normal distribution curve are called bell curves as the line of the graph looks like a bell

The mathematical name for this bell curve is "normal distribution" where the measurements tend to cluster around the mean and tapper off at the edges.

So the curve of the two graphs above would look like:

Bell curve of class 3 test scores

Bell curve of class 4 test scores

This form of expression using the words "standard deviation" only works for this normal distribution or bell curve. It would not work for any curve that was skewed as:

Two graphs one with the data skewed to the right and the other with the data skewed to the left

Or

Graph with the data jumbeled up all over the place has a wavey line

So taking the two graphs above and expressing them as a curve:

The 68% (1 standard deviation) of Class 3 = 35

The 68% (1 standard deviation) of Class 4 = 15

As follows graphically:

Mean and 1 Standard deviation of class 3 test scores shown on a bell curveMean and 1 standard deviation of class 4 test scores shown on a bell curve

So if you compare

Class 3 mean = 85 and 1 SD (standard deviation) = 35

Class 4 mean = 85 and 1 SD (standard deviation) = 15

You can tell just from the numbers that class 4 has a much more concentrated cluster around the middle than class 3.

The teacher in class 3 may need some extra help for the wide spread of distribution of kids in her class.

More help for the not so highly scoring kids and even more help perhaps for the really smart kids who need extra help to push them on. The graph is so diverse they certainly can't be taught in the same way. Where as class 4 the teacher may have a better chance to bring all the kids through together. 

All this just from the numbers.

 

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