If the standard deviation of a probability distribution is 4 what is the variance

Deviation just means how far from the normal

Standard Deviation

The Standard Deviation is a measure of how spread out numbers are.

Its symbol is σ (the greek letter sigma)

The formula is easy: it is the square root of the Variance. So now you ask, "What is the Variance?"

Variance

The Variance is defined as:

The average of the squared differences from the Mean.

To calculate the variance follow these steps:

• Work out the Mean (the simple average of the numbers)
• Then for each number: subtract the Mean and square the result (the squared difference).
• Then work out the average of those squared differences. (Why Square?)

Example

You and your friends have just measured the heights of your dogs (in millimeters):

The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm.

Find out the Mean, the Variance, and the Standard Deviation.

Your first step is to find the Mean:

Answer:

so the mean (average) height is 394 mm. Let's plot this on the chart:

Now we calculate each dog's difference from the Mean:

To calculate the Variance, take each difference, square it, and then average the result:

So the Variance is 21,704

And the Standard Deviation is just the square root of Variance, so:

And the good thing about the Standard Deviation is that it is useful. Now we can show which heights are within one Standard Deviation (147mm) of the Mean:

So, using the Standard Deviation we have a "standard" way of knowing what is normal, and what is extra large or extra small.

Rottweilers are tall dogs. And Dachshunds are a bit short, right?

Using

We can expect about 68% of values to be within plus-or-minus 1 standard deviation.

Read Standard Normal Distribution to learn more.

Also try the Standard Deviation Calculator.

But ... there is a small change with Sample Data

Our example has been for a Population (the 5 dogs are the only dogs we are interested in).

But if the data is a Sample (a selection taken from a bigger Population), then the calculation changes!

When you have "N" data values that are:

• The Population: divide by N when calculating Variance (like we did)
• A Sample: divide by N-1 when calculating Variance

All other calculations stay the same, including how we calculated the mean.

Example: if our 5 dogs are just a sample of a bigger population of dogs, we divide by 4 instead of 5 like this:

Sample Variance = 108,520 / 4 = 27,130

Sample Standard Deviation = √27,130 = 165 (to the nearest mm)

Think of it as a "correction" when your data is only a sample.

Formulas

Here are the two formulas, explained at Standard Deviation Formulas if you want to know more:

Looks complicated, but the important change is to
divide by N-1 (instead of N) when calculating a Sample Standard Deviation.

*Footnote: Why square the differences?

If we just add up the differences from the mean ... the negatives cancel the positives:

4 + 4 − 4 − 44 = 0

So that won't work. How about we use absolute values?

|4| + |4| + |−4| + |−4|4 = 4 + 4 + 4 + 4 4 = 4

That looks good (and is the Mean Deviation), but what about this case:

|7| + |1| + |−6| + |−2|4 = 7 + 1 + 6 + 2 4 = 4

Oh No! It also gives a value of 4, Even though the differences are more spread out.

So let us try squaring each difference (and taking the square root at the end):

That is nice! The Standard Deviation is bigger when the differences are more spread out ... just what we want.

In fact this method is a similar idea to distance between points, just applied in a different way.

And it is easier to use algebra on squares and square roots than absolute values, which makes the standard deviation easy to use in other areas of mathematics.

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