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Standard Deviation: Definition, Properties, & Calculations

Standard deviation is a significant statistical term. It plays a key role in measuring the dispersion of data values in different industrial sectors and population departments as well as in insurance companies and the engineering sector.

The standard deviation shows how many variations there are from the mean value and gives an idea of the shape of the distribution. Standard deviation and variance both terms are interconnected to each other similar to the square and square root of a number.

How much data values spread about the mean is measured using the concept of variance. It combines all the data values in a data set to give a measure of the variation of the data values around the average. SD measures how far, on average, each data value differs from the average (mean)

In this article, we will discuss an important statistical term the standard deviation. We will address its definition and its mathematical formula with examples.

Standard Deviation: Definition, Properties, & Calculations

Standard Deviation

When we take the square root of the variance, we get the value which is called the standard deviation (SD). It measures the spread of the data, showing how much the individual data points turn, on average, from the mean of the dataset. It has the measuring units the same as the units of the data set/point.  It is the square root of the sum of the squared distances from the mean of data values.

Since standard deviation and variance are interlinked terms and the variance is calculated by taking the mean (average) of the squares of difference b/w each data value and the mean, by taking the square root of variance end up with a number that is more easily comparable to the original data in the list.

Mathematically,

σ = √[∑(x−x̄)2 /n]

If we take a square of the above expression

i.e. σ2 = (∑x−x̄)2/n).

Which represents the term variance that gives a mean average of the squared distance b/w each data point and the mean.

Here represents the average or mean of the given data values. Now we will elaborate on some important properties of the standard deviation that play a vital role in getting insights from the data sets.

Properties of Standard Deviation

  • SD is only used to measure the spread or dispersion of a data set around the mean (average).
  • Standard deviation is never negative because it is an average of the squared distances (x−x̄)2.
  • SD is sensitive to an outlier value in the given data sets/pints. Even a single outlier can elevate (raise) the SD value and in turn, wring (deform) the picture of the expansion or dispersion.
  • For data with approximately the same mean, the greater the spread, the greater the standard deviation.
  • A small value of standard deviation (SD) shows that the data values are closer to the mean than the large value of standard deviation.

Steps to Compute Standard Deviation

It is very important to understand how to calculate standard deviation of the given data. Here we will elaborate to calculate SD from the given data.

  • Step 1: First of all, find out the mean (average) of the given data which is equivalent to some of the observations divided by the number of observations
  • x̄ = ∑x /n
  • Step 2: Compute the difference (x – x̄) for each term of the given data set.
  • Step 3: Square all the values computed in step 2 i.e. (x – x̄)2.
  • Step 4: Now sum up all the values computed in step 3 i.e. ∑ (x – x̄)2.
  • Step 5: Divide this cumulative value (∑ (x – x̄)2) by the total number of observations i.e. (∑x−x̄)2/n) which is the variance (σ2).
  • Step 6: In this step, we will take the square root of the variance i.e. σ = √[∑(x−x̄)2 /n] which will be the required result.

Calculations:

Now let us see some examples to comprehend the term standard deviation precisely.

Example 1:

Calculate the standard deviation for the given data.

Solution:

Step 1: Write down the given data values in arranged form

1, 1, 2, 2, 3, 3, 4, 5, 6

Step 2: Calculate the average (mean) of the given data

x̄ = ∑x /n

x̄ = (1+1+2+2+3+3+4+5+6) / 9

x̄ = 27/9

x̄ = 3

Step 3: Now we will perform steps 2 and 3 as illustrated in the following table.

x(x – x̄)(x – x̄)2
1-24
1-24
2-11
2-11
300
300
411
524
639

Step 4: Perform steps 4 and 5.

σ2 = (∑x−x̄)2 /n)

σ2 = (4+4+1+1+0+0+1+4+9)/ 9

σ2 = 24/9

σ2 = 2.67 (Variance is the sum of the squared distances from the mean, divided by the number of data values.

Step 5: We take the square root on both sides.

σ = + 1.67 (Standard deviation can never be negative).

Understanding Standard Form in Mathematics

Example 2:

Calculate the standard deviation for the given data

4, 3, 1, 2, 6, 3, 2, 5, 1, 1, 2, 6

Solution:

Step 1: Write down the given data values in arranged from

1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 6

Step 2: Find out the mean (average) of the given data

x̄ = (1+1+1+2+2+2+3+3+4+5+6+6)/ 12

x̄ = 36/12

x̄ = 3

Step 3: Now we will perform steps 2 and 3 as illustrated in the following table.

x(x – x̄)(x – x̄)2
1-24
1-24
1-24
2-11
2-11
2-11
300
300
411
524
639
639

Step 4: Perform steps 4 and 5.

σ2 = (∑x−x̄)2/n)

σ2 = (4+4+4+1+1+1+0+0+1+4+9+9)/ 12

σ2 = 38/12

σ2 = 3.17

Step 5: We take the square root on both sides.

σ = 1.78 which is the required result.

Wrap Up:

In this article, we have elaborated on the statistical term standard deviation, its definition, and the formula to find out standard deviation. We have also discussed its important properties that are very useful for the computation of statistical problems and the study of statistical theory.

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