standard-deviation-sd-definition-formulas-and-examples |

Standard Deviation (SD) is a
significant statistical term. It plays a crucial role to measure the dispersion
of datasets in real-life problems in several industrial sectors, population departments,
insurance companies, engineering, etc.

The standard deviation shows how many
variations there are from the mean 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.

The variance combines all the data
values in a data set to give a measure of a data’s spread about its mean. SD measures
how far, on average, each data value differs from the average (mean). In this
blog, we will discuss the term standard deviation, its formula, some useful
properties, and examples.

## What is a Standard Deviation (SD):

When we take the square root of the variance,
we get the value called standard deviation (SD). SD is the mensuration of how
spread out the data is. SD has the measuring units the same as the units of the
data set/point. SD is the square root of
the sum of the squared distances from the mean of data values.

## Formula of Standard Deviation:

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, so 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,

**Ïƒ**** = √[∑(x−x̄) ^{2} /n]**

If we take a
square of the above expression i.e. Ïƒ^{2} = (∑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 x̄ is
the average (mean) of the data sets. Now we will elaborate on some important
properties of the standard deviation that play a vital role to get 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 low
standard deviation (SD) shows that the data points tend to be closer to the
mean than a larger standard deviation.

## Method to Compute SD:

It is very important to apprehend how
to calculate the standard deviation of the given data as this term is very
frequently used in our life. 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 the 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: Dived this cumulative value
(∑(x - x̄)2) by the total number of observations i.e. (∑x−x̄)2/n) which is the
variance (Ïƒ2).

Step 6: In the last step, you only need
to take the square root of the variance i.e. Ïƒ = √[∑(x−x̄)2
/n] which will be the required result.

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

## Example:

**Example 1: **

Calculate the standard
deviation for the given data

2, 1, 3, 5, 0, 4, 1, 3, 6,

**Solution: **

**Step 1:** Write down the given
data 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) |

1 |
-2 |
4 |

1 |
-2 |
4 |

2 |
-1 |
1 |

2 |
-1 |
1 |

3 |
0 |
0 |

3 |
0 |
0 |

4 |
1 |
1 |

5 |
2 |
4 |

6 |
3 |
9 |

**Step 4:** Perform steps 4
and 5.

Ïƒ^{2} = (∑x−x̄)^{2}/n)

Ïƒ^{2} = (4+4+1+1+0+0+1+4+9)/ 9

Ïƒ^{2} = 24/9

Ïƒ^{2} = 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.

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

**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 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) |

1 |
-2 |
4 |

1 |
-2 |
4 |

1 |
-2 |
4 |

2 |
-1 |
1 |

2 |
-1 |
1 |

2 |
-1 |
1 |

3 |
0 |
0 |

3 |
0 |
0 |

4 |
1 |
1 |

5 |
2 |
4 |

6 |
3 |
9 |

6 |
3 |
9 |

**Step 4:** Perform steps 4
and 5.

Ïƒ^{2} = (∑x−x̄)^{2}/n)

Ïƒ^{2} = (4+4+4+1+1+1+0+0+1+4+9+9)/ 12

Ïƒ^{2} = 38/12

Ïƒ^{2} = 3.17

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

Ïƒ = 1.78 which is the required
result.

# Summary:

In this
blog, we have elaborated on the statistical term standard deviation, its
definition, and the formula to find out SD. 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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