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Normal Distribution

Data can be "distributed" (spread out) in different ways.

It can be spread out
more on the left
Graph with data points clustered on the right and a tail on the left

Skewed left

Or more on the right

Graph with data points clustered on the left and a tail on the right

Skewed right

Or it can be all jumbled up

Scatter plot showing data points with no clear pattern or bias

But in many cases the data tends to be around a central value, with no bias left or right, and it gets close to a "Normal Distribution" like this:

Smooth bell curve overlaying a yellow bar histogram

The blue curve is a Normal Distribution.
The yellow histogram shows some data that
follows it closely, but not perfectly (which is usual).

It is often called a "Bell Curve"
because it looks like a bell.

Many things closely follow a Normal Distribution:

heights of people
size of manufactured products
measurement errors
blood pressure
test scores

We say the data is "normally distributed":

normal distribution with mean median mode at center

The Normal Distribution has:

mean = median = mode
symmetry about the center
50% of values less than the mean
and 50% greater than the mean

Most data isn't perfectly normal, but the normal distribution still helps us make sense of it!

Quincunx

You can see a normal distribution being created by random chance!

It is called the Quincunx and it is an amazing machine. Have a play with it!

Standard Deviations

The Standard Deviation is a measure of how spread out numbers are (read that page for details on how to calculate it).

When we calculate the standard deviation we find that approximately:

Normal distribution bell curve with 68% highlighted within 1 standard deviation

68% of values are within
1 standard deviation of the mean

Normal distribution bell curve with 95% highlighted within 2 standard deviations

95% of values are within
2 standard deviations of the mean

Normal distribution bell curve with 99.7% highlighted within 3 standard deviations

99.7% of values are within
3 standard deviations of the mean

This is called the 68-95-99.7 rule

Example: 95% of students at school are between 1.1m and 1.7m tall.

Assuming this data is normally distributed can you calculate the mean and standard deviation?

The mean is halfway between 1.1m and 1.7m:

Mean = (1.1m + 1.7m) / 2 = 1.4m

95% is 2 standard deviations either side of the mean (a total of 4 standard deviations) so:

1 standard deviation	= (1.7m-1.1m) / 4
	= 0.6m / 4
	= 0.15m

And this is the result:
Bell curve with mean 1.4m and 95% of data between 1.1m and 1.7m

It is good to know the standard deviation, because (for data that's approximately normal) we can say that any value is:

likely to be within 1 standard deviation (68 out of 100 should be)
very likely to be within 2 standard deviations (95 out of 100 should be)
almost certainly within 3 standard deviations (997 out of 1000 should be)

Standard Scores

The number of standard deviations from the mean is also called the "Standard Score" or "z-score". Get used to those words!

σ (sigma) is the standard deviation
z is the standard score (how many standard deviations from the mean)

Example: In that same school one of your friends is 1.85m tall

Bell curve with mean 1.4m and 95% of data between 1.1m and 1.7m

You can see on the bell curve that 1.85m is 3 standard deviations from the mean of 1.4, so:

Your friend's height has a "z-score" of 3.0

It is also possible to calculate how many standard deviations 1.85 is from the mean

How far is 1.85 from the mean?

It is 1.85 - 1.4 = 0.45m from the mean

How many standard deviations is that? The standard deviation is 0.15m, so:

0.45m / 0.15m = 3 standard deviations

To convert a value to a Standard Score ("z-score"):

first subtract the mean,
then divide by the Standard Deviation

And doing that's called "Standardizing":

Flowchart showing a specific normal curve being transformed into the standard normal curve

We can take any Normal Distribution and convert it to The Standard Normal Distribution.

Example: Travel Time

A survey of daily travel time had these results (in minutes):

26, 33, 65, 28, 34, 55, 25, 44, 50, 36, 26, 37, 43, 62, 35, 38, 45, 32, 28, 34

The Mean is 38.8 minutes, and the Standard Deviation is 11.4 minutes (you can copy and paste the values into the Standard Deviation Calculator if you want).

Convert the values to z-scores ("standard scores").

To convert 26:

first subtract the mean: 26 − 38.8 = −12.8,

then divide by the Standard Deviation: −12.8/11.4 = −1.12

So 26 is −1.12 Standard Deviations from the Mean

Here are the first three conversions

Original Value	Calculation	Standard Score (z-score)
26	(26-38.8) / 11.4 =	−1.12
33	(33-38.8) / 11.4 =	−0.51
65	(65-38.8) / 11.4 =	+2.30
...	...	...

And here they are graphically:

Travel time z-scores plotted at -1.12, -0.51, and +2.30 on a standard normal curve

You can calculate the rest of the z-scores yourself!

The z-score formula that we have been using is:

z = x − μσ

z is the "z-score" (Standard Score)
x is the value to be standardized
μ ("mu") is the mean
σ ("sigma") is the standard deviation

And this is how to use it:

Example: Travel Time (continued)

Here are the first three conversions using the "z-score formula":

z = x − μσ

μ = 38.8
σ = 11.4

x	x − μσ	z (z-score)
26	26 − 38.811.4	= −1.12
33	33 − 38.811.4	= −0.51
65	65 − 38.811.4	= +2.30
...	...	...

The exact calculations we did before, just following the formula.

Why Standardize ... ?

It can help us make decisions about our data.

Example: Professor Willoughby is marking a test.

Here are the students' results (out of 60 points):

20, 15, 26, 32, 18, 28, 35, 14, 26, 22, 17

Most students didn't even get 30 out of 60, and most will fail.

The test must have been really hard, so the Prof decides to Standardize all the scores and only fail people more than 1 standard deviation below the mean.

The Mean is 23, and the Standard Deviation is 6.6, and these are the Standard Scores:

-0.45, -1.21, 0.45, 1.36, -0.76, 0.76, 1.82, -1.36, 0.45, -0.15, -0.91

Now only 2 students will fail (the ones lower than −1 standard deviation)

Much fairer!

It also makes life easier because we only need one table (the Standard Normal Distribution Table), rather than doing calculations individually for each value of mean and standard deviation.

In More Detail

Here's the Standard Normal Distribution with percentages for every half of a standard deviation, and cumulative percentages:

Standard normal distribution showing percentage areas for each 0.5 standard deviation interval

Example: Your score in a recent test was 0.5 standard deviations above the average, how many people scored lower than you did?

Between 0 and 0.5 is 19.1%
Less than 0 is 50% (left half of the curve)

So the total less than you is:

50% + 19.1% = 69.1%

In theory 69.1% scored less than you did (but with real data the percentage may be different)

Kitchen scale weighing a 1 kg bag of sugar

A Practical Example: Your company packages sugar in 1 kg bags.

When you weigh a sample of bags you get these results:

1007g, 1032g, 1002g, 983g, 1004g, ... (a hundred measurements)
Mean = 1010g
Standard Deviation = 20g

Some values are less than 1000g ... can you fix that?

The normal distribution of your measurements looks like this:

Normal curve with mean 1010g showing 31% shaded below 1000g

31% of the bags are less than 1000g,
which is cheating the customer!

It is a random thing, so we can't stop bags having less than 1000g, but we can try to reduce it a lot.

Let's adjust the machine so that 1000g is ...

... at −3 standard deviations:

From the big bell curve above we see that only 0.1% are less. But maybe that's too small.

... at −2.5 standard deviations:

Below 3 is 0.1% and between 3 and 2.5 standard deviations is 0.5%, together that's 0.1% + 0.5% = 0.6% (a good choice I think)

So let's adjust the machine to have 1000g at −2.5 standard deviations from the mean.

Now, we can adjust it to:

increase the amount of sugar in each bag (which changes the mean), or
make it more accurate (which reduces the standard deviation)

Let's try both.

Adjust the mean amount in each bag

Shifted normal curve with mean 1050g and 0.6% shaded below 1000g

The standard deviation is 20g, and we need 2.5 of them:

2.5 × 20g = 50g

So the machine should average 1050g, like this:

Adjust the accuracy of the machine

Narrow normal curve with standard deviation 4g and 0.6% shaded below 1000g

Or we can keep the same mean (of 1010g), but then we need 2.5 standard deviations to be equal to 10g:

10g / 2.5 = 4g

So the standard deviation should be 4g, like this:

(We hope the machine is that accurate!)

Or perhaps we could have some combination of better accuracy and slightly larger average size, I will leave that up to you!

More Accurate Values ...

Use the Standard Normal Distribution Table when you want more accurate values.

2619, 2620, 2621, 2622, 2623, 2624, 2625, 2626, 3844, 3845

Standard Deviation Standard Deviation Calculator Standard Normal Distribution Table Quincunx Normal Approximation to Binomial Distribution Probability and Statistics Index