In Statistics, for many of the computational problems, we have to analyze the given data. Standard Deviation is a statistical method used to find or measure the amount of scatteredness around an average of data under observation. Dispersion means the difference between the original values and average values.

The more this dispersion in the given data, the bigger the value of the standard deviation. Standard Deviation may be symbolized by sigma* ***σ** (a Greek letter) for the population **S.D**. Standard Deviation is the most commonly used measure of dispersion, which is based on all values given in data.

It means if we change anyone’s value, the whole value of standard deviation will be affected. In many advanced statistical problems, the concept of standard deviation is widely used. In this article, we will discuss standard deviation; formulas depending upon the nature of data whether grouped data or ungrouped data, properties, key points, and examples.

## What is Standard Deviation?

In statistical calculations, the measurement of variability in data from its mean is known as the standard deviation (SD). It tells us the degree of deviation from the data’s average (mean).

__For Group data__

**Standard Deviation = √[(∑FX ^{2}/∑F) – (∑FX/∑F)^{2}]**

__For Ungroup data__

**Standard Deviation = √[(∑X ^{2}/n) – (∑X/n)^{2}]**

### Key points:

- Standard Deviation measures the dispersion of a set of data relative to its mean.
- If the variance of given data is given, then the standard deviation will be calculated easily by taking the square root of variance.
- In Finance, the standard deviation is often used to measure the relative riskiness of assets.
- As a downside, the standard deviation evaluates all uncertainty as a risk even when it’s in the investor’s favor. For example, to calculate the average returns of investors, we have to calculate its standard deviation.
- A low standard deviation shows that the value of under-observation data tends to be close to the mean. On the other hand, a high standard deviation shows that the values are spread out over a bigger range.

## Properties of Standard Deviation

Standard deviation is quite useful in many of the problems in Statistics to analyze the given data. To have an easy approach to different problems, we should have a look at its properties.

**i)**The standard deviation of any constant is always equal to zero. If “**a**” is a constant, then**S.D.(a) = 0****ii)**The standard deviation is independent of origin.**S.D. (x ± a) = S.D.(x)****iii)**When all the values are multiplied with a constant, the standard deviation will be multiplied by the constant.

**S.D. (ax) = a. S.D. (x)**And

S.D. (x/a) = (1/a) . S.D. (x)

S.D. (x/a) = (1/a) . S.D. (x)

**iv) **The standard deviation of the sum or difference of two independent variables is equal to the sum of their respective standard deviation.

**S.D. ( x + y ) = S.D. (x) + S.D. (y) **And **S.D. ( x – y ) = S.D. (x) + S.D. (y)**

### Examples

Below are a few examples of finding standard deviation for grouped and ungrouped data.

**Example 1: (Group data)**In a Middle School, the weights of 400 students are given below:

Weights | 41-50 | 51-60 | 61-70 | 71-80 | 81-90 | 91-100 |

Freq | 30 | 36 | 143 | 104 | 73 | 14 |

Find its standard deviation.

**Solution:**

The formula for group data for evaluating standard deviation is given below

Standard Deviation = √[(∑FX^{2}/∑F) – (∑FX/∑F)^{2}]

Weights | Frequency | X | X^{2} | FX | FX^{2} |

41-50 | 30 | 45.5 | 2070.25 | 1365 | 62107.5 |

51-60 | 36 | 55.5 | 3080.25 | 1998 | 110889 |

61-70 | 143 | 65.5 | 4290.25 | 9366.5 | 613505.75 |

71-80 | 104 | 75.5 | 5700.25 | 7852 | 592826 |

81-90 | 73 | 85.5 | 7310.25 | 6241.5 | 533648.25 |

91-100 | 14 | 95.5 | 9120.25 | 1337 | 127683.5 |

∑F = 400 | | | ∑FX = 28160 | ∑FX^{2} = 2040660 |

Standard Deviation = √ [(∑FX^{2}/∑F) – (∑FX/∑F)^{2}]

= √ [ (2040660 / 400) – (28160 / 400 )^{2}]

= √ [ 5101.65–4956.16]

= √ 145.49

**S.D = 12.06**

**Example 2: (Ungroup data)**

The profit of some articles in a market in USD is given as 63,52,32,87,36,23,75,72,83,23,78,97,31,43. Calculate its Standard Deviation.

**Solution:**

Standard Deviation = √[(∑X^{2}/n) – (∑X/n)^{2}]

__Step 1:__

∑ (X) = (63 + 52 + 32 + 87 + 36 + 23 + 75 + 72 + 83 + 23 + 78 + 97 + 31+ 43)

∑ (X) = 795

∑ (X^{2}) = (63^{2} + 52^{2} + 32^{2} + 87^{2} + 36^{2} + 23^{2} + 75^{2} + 72^{2} + 83^{2} + 23^{2} + 78^{2} + 97^{2} + 31^{2}+ 43^{2})

∑ (X^{2}) = 53621

__Step 2:__

Standard Deviation = √ [(53621/14) – (795/14)^{2}]

= √ [ 3830.07 – 3224.61 ]

= √605.46

**S.D. = 24.61**

## Summary

In this article, we have discussed the concept of Standard deviation precisely in different directions. As it has been clarified to know the variation or measure of the dispersion of given data, we easily can find it through standard deviation.

Depending upon the nature of the data, two formulas are discussed above in this article. By understanding its examples, we can solve all statistical computations easily. Many of the basic properties of Standard deviation have also been mentioned above through which many complex or infinite data sets are analyzed easily.

## FAQ’s

### What is the use of Standard Deviation in real life?

**Ans: **Standard deviation is utilized in many fields like statistics, finance, business, and many others. By knowing it, we can easily acknowledge the distribution of the data set. In quality control, it tells us about the variability in manufacturing processes.

### Can the Standard Deviation of data be negative?

**Ans: **No, the standard deviation of data can never be negative. The reason is that its value is computed after taking the square root of the variance and the square root is only possible when the radicand of it is in a positive integer. However, the standard deviation may be zero, if the provided dataset is constant.

### What are the maximum and minimum outputs of standard deviation?

**Ans: **The minimum output of standard deviation is zero but as far as its maximum output is concerned, it always depends upon provided data. If the variability in the provided data is high, then the standard deviation will also be high or maximum.