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.
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.
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 n1. We call this the variance.
Variance =
(x – x)^{2}
n – 1
Why is the denominator n1 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 n1 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.
$$ Sample Standard Deviation = s = sqrt { frac {Σ (x – bar x) ^ {2}} {n – 1}} $$
$$ Population Standard Deviation = σ = sqrt { frac {Σ (x – µ) ^ {2}} {N}} $$

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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