The standard deviation (SD) indicates how far the values ​​of a data set are from the mean.

If the standard deviation is large, the mean values ​​are far from the mean.

On the other hand, if the standard deviation is small, the mean values ​​are close to the mean value.

To see what we mean by “near or far from the mean,” consider the following data, which shows the grades of 5 students.

5 20 40 60 100

When you have calculated the mean and SD for this data set, you will find that the mean is 45 and the SD is about 37.

The following graphic shows the deviations from the mean. As you can see, the deviations shown with green lines are large.

Standard deviation

This means that, on average, some grades are far from the mean, either below or above.

This time, however, consider the following notes.

42 45 50 38 50

This time the mean is again 45, but the SD is only 5.2

The deviations are small this time.

Standard deviation

How to calculate the standard deviation

Since the SD tries to find out how far the values ​​deviate from the mean, it makes sense to calculate the deviations from the mean for all values.

Let’s see how the SD was determined for the following notes.

42 45 50 38 50

To get the deviation from the mean for a single value, simply subtract the mean from that value.

The mean is 45.

42 – 45 = -3

45 – 45 = 0

50 – 45 = 5

38 – 45 = -7

50 – 45 = 5

It also makes sense to add -3, 0, 5, -7 and 5 together and then divide the result by 4 to get the “average” SD

-3 + 0 + 5 + -7 + 5 = 0 and 0 divided by 4 = 0.

Since there are 5 numbers, why do we divide by 4 instead of 5?

The reason is explained in the last section below.

We don’t get anything useful if we add -3, 0, 5, -7, and 5. To avoid 0, we can square the values ​​before adding them up.

(-3)2 = 9

(0)2 = 0

(5)2 = 25

(-7)2 = 49

(5)2 = 25

9 + 0 + 25 + 49 + 25 = 108

Dividing 108 by 4 gives the variance or sample Deviation.

Deviation = 108/4 = 27

Since we had to square all values ​​to avoid zero, we need to take the square root of 27 to get the standard deviation.

Standard deviation = square root of 27 = 5.19 or 5.20

How to derive the standard deviation formula

To find the formula for the standard deviation we just need to generalize what we did above. Here are the steps.

Step 1:

Find the mean or x

Step 2:

Subtract the mean from the values. Let x be all values ​​and x the mean.

We get x- x

Step 3:

Square each of the differences we found in step 2.

(x- x)2

Step 4:

Add all of the squares you found in step 3

(x – x)2

Step # 5:

Let n be the number of the value. Remember, in our example above, we divided by 4 instead of 5.

If n is the number of values, divide the above sum by n-1. We call this the variance.

Variance =

(x – x)2

/
n – 1

Why is the denominator n-1 instead of n?

  • The main reason we’re looking for sample variance is to estimate population variance. It has been observed that a denominator of n-1 does a better job of estimating population variance.
  • Division by n tends to underestimate the population variance. In other words, dividing by n causes the population variance to be smaller than it actually is.


Step # 6:

Remember that in step 3 we had to square the differences. Hence, we need to take the square root of the variance to get the standard deviation.

$$ Sample Standard Deviation = s = sqrt { frac {Σ (x – bar x) ^ {2}} {n – 1}} $$

$$ Population Standard Deviation = σ = sqrt { frac {Σ (x – µ) ^ {2}} {N}} $$


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  1. Standard deviation

    May 28, 21, 4:49 a.m.

    What is standard deviation? Crystal clear explanation of the standard deviation. Learn to calculate it and understand how the formula was derived.

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