The collection and representation of data in the study of statistics are demonstrated using the idea of the measures of dispersion such as variance and standard deviation of ungrouped data.

This variation in the information collected from the research will help in a deeper comprehension of this concept.

The interpretation and representation of the data under study are observed and understood by employing the concepts of variance and standard deviation.

Variance can never be negative and it can only ever be zero or positive. If the data values under consideration are equal, the variance can have a zero value.

In this comprehensive discussion, we will explain how to calculate standard deviation and variance for ungrouped data.

We will address their relations for population and sample variance and standard deviation in detail. We will also solve some examples for better understanding.

**What is Variance?**

A measure of dispersion or variation that elaborates on the degree of variability among the values in a collection of data is known as variance.

It is employed to calculate the distance between the mean and every number in the collection.

Variance is a word used by traders and analysts to describe market security and volatility. There are two different kinds of variation: sample variance and population variance. we will explain these terms.

**Population Variance:**

The concept of population variance is applied to determine the variance for a whole population. We are to do the important work to collect information from every member of the population to obtain a precise estimation.

The mathematical relation to determining the population variance of ungrouped data is given below:

σ2 = ∑ (xi – μ)2 / N

σ2 represents the population variance for ungrouped data and ∑ is the sigma sign. It is important to note that in the above formula, xi is the relevant observation or value.

This formula helps to determine the population variance for the problems of the raw distribution data.

**Sample Variance:**

The concept of sample variance is applied to determine the variation for a certain sample. Estimates or conclusions on the population variance can also be determined using the sample variance.

The mathematical relation to determine the sample variance of ungrouped data is given below:

S2 = ∑ (xi – x̅)2 / n – 1

S2 represents the sample variance for grouped data and ∑ is the sima sign. It is important to note that n is the number of data values or information in the above-written mathematical formula.

This formula helps to determine the sample variance for the problems of the raw distribution data.

**Standard Deviation (SD):**

The degree of variance or dispersion of a collection of collective data is measured by the standard deviation.

A small value of the standard deviation number signifies that the values are quite close to the selected mean. Values are dispersed over a large range when the standard deviation is high.

Standard deviation is also of two different kinds population standard deviation and sample standard deviation.

Whether the data is interpreted as a sample that represents a broader population or as a population of its own determines the formula, we use to calculate the standard deviation.

Now we will explain both of these terms below.

**Population Standard Deviation (PSD):**

The standard deviation for the entire population is found by using the population standard deviation concept.

For a precise estimation of the value of standard deviation, we need to collect data from every member of the population.

We divide the data by the total number of data points (N) if the data is to be regarded as a population in and of itself.

σ = √ (∑ (xi – μ)2 / N)

**Sample Standard Deviation (SSD):**

One way to ascertain the variation for a given sample is to use the concept of the sample standard deviation.

We can additionally get approximations or inferences about the standard deviation of the entire population from the sample standard deviation.

If the data represents a sample taken from a larger population, then we divide the sample size by one less than the total number of data points (n – 1).

S = √ [∑ (xi – x̅)2 / (n -1)]

Example Section:

Below are a few examples of finding standard deviation and variance.

Example:

Given the scores of the students in the following table:

Student | Asad | Usman | Umair | Sami | Bilal | Ayyan | Zain | Niaz | Umer | Moiz | Ali | Uzair |

Score Xi | 7 | 11 | 8 | 8 | 19 | 15 | 7 | 9 | 9 | 20 | 17 | 14 |

Determine what will be the values of SD and variance.

Solution:

Step 1: Determine the average of the scores given above in the table:

x̅ = μ = (7 + 11 + 8 + 8 + 19 + 15 + 7 + 9 + 9 + 20 + 17 + 14) 12

x̅ = μ = (144) / 12

x̅ = μ = 12

Step 2: Now we will compute the necessary computations performed in the following table:

xi | xi – x̅ = xi – μ | (xi – x̅)2 = (xi – μ)2 |
---|---|---|

7 | -5 | 25 |

11 | -1 | 1 |

8 | -4 | 16 |

8 | -4 | 16 |

19 | 7 | 49 |

15 | 3 | 9 |

7 | -5 | 25 |

9 | -3 | 9 |

9 | -3 | 9 |

20 | 8 | 64 |

17 | 5 | 25 |

14 | 2 | 4 |

∑ | 252 |

Step 3: Now the mathematical relation for population variance:

**σ2 = ∑ (xi – μ)2 / N**

Put the relevant values in the above formula:

σ2 = (252) / 12

σ2 = 21 Ans.

Now calculation for sample variance:

Formula:

**S2 = ∑ (xi – x̅)2 / n – 1**

S2 = (252) / ((12 – 1)

S2 = (252) / 11

S2 = 22.91 Ans.

Step 4: Now the mathematical relation for population standard deviation:

**σ = √ (∑ (xi – μ)2 / N)**

σ = √ (21)

σ = 4.58 Ans.

Now calculation for sample standard deviation:

Formula:

**S = √ [∑ (xi – x̅)2 / (n -1)]**

S = √ (22.91)

S = 4.7863 Ans.

Example 2:

For the following data given in the table, determine the values for SD and variance.

City | A | B | C | D | E |

Distance Xi | 6 | 7 | 8 | 9 | 10 |

Solution:

Step 1: Determine the average of the scores given above in the table:

x̅ = μ = (6 + 7 + 8 + 9 + 10) / 5

x̅ = μ = (40) / 5

x̅ = μ = 8

Step 2: Now we will compute the necessary computations performed in the following table:

Xi | xi – x̅ = xi – μ | (xi – x̅)2 = (xi – μ)2 |
---|---|---|

6 | -2 | 4 |

7 | -1 | 1 |

8 | 0 | 0 |

9 | 1 | 1 |

10 | 2 | 4 |

∑ | 10 |

Step 3: Now the mathematical relation for population variance:

**σ2 = ∑ (xi – μ)2 / N**

Put the relevant values in the above formula:

σ2 = (10) / 5

**σ2 = 2 Ans.**

Now calculation for sample variance:

Formula:

**S2 = ∑ (xi – x̅)2 / n – 1**

S2 = (10) / ((5 – 1)

S2 = (10) / 4

**S2 = 2.5 Ans.**

Step 4: Now the mathematical relation for population standard deviation:

**σ = √ (∑ (xi – μ)2 / N)**

σ = √ (2)

**σ = 1.4142 Ans.**

Now calculation for sample standard deviation:

Formula:

**S = √ [∑ (xi – x̅)2 / (n -1)]**

S = √ (2.5)

**S = 1.5811 Ans.**

**Summary:**

In this comprehensive discussion, we have elaborated on the concept of the standard deviation and the variance. We discussed in detail how to compute the SD and variance for ungrouped data with the help of the solved example.

I hope that by reading this article you will understand the important terms of SD and variance precisely. It will also enable you to solve the problems related to these terms.

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