# Measures of the Spread of Data

Summarising the dataset can help you to understand the data, especially when the dataset is too large. A**Measures of Central Tendency**of a dataset by itself is not enough, though, to describe a distribution. The

**Measures of the Spread**, sometimes also called a measure of dispersion, of dataset describe you how extreme the values in the dataset are. It summarise the dataset in a way that explain how scattered the values are and how much they differ from the mean value.

There are several basic measures of spread used in statistics. The most common are:

- Range
- Inter-Quartile Range (IQR)
- Variance
- Standard Deviation

## Range

The**Range**is Define and calculate the range of a dataset. It is the difference between the

**smallest**value in a dataset (Minimum) and the

**largest**one (Maximum).

**Range = maximum - minimum**

Suppose you have a dataset of some values:

12, 48, 32, 21, 32, 36, 54, 21, 78, 32, 18, 94.

*Minimum value is*:

**12**

*Maximum value is*:

**94**

*Range of dataset is*:

**94 - 12 = 82**

## Quartile and Inter-Quartile Range (IQR)

### Quartile

Quartiles are the values that divide a dataset into**quarters**.

### Inter-Quartile Range (IQR)

The IQR (**Inter-Quartile Range**) is a measure of variability, based on dividing a dataset into quartiles. First Quartile (Quartile-1) is denoted by

**Q1**known as the lower quartile, the second Quartile (Quartile-2) is denoted by

**Q2**and the third Quartile (Quartile-3) is denoted by

**Q3**known as the upper quartile. The interquartile range is found by subtracting the

**Q1**value from the

**Q3**value.

**IQR = Q3 - Q1**

### How to find Inter-Quartile Range?

Suppose you have a dataset:

60, 110, 30, 10, 40, 20, 100, 90, 70, 80, 50.

Put the numbers in order.

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110.

### Find the median of dataset

Total eleven numbers in the dataset. So, the**median**is sixth value:

*Median value is*:

**60**

Separate the numbers above and below the median.

(10, 20, 30, 40, 50) 60 (70, 80, 90, 100, 110)

In the above case Q2 is **60**. So, find the Q1 and Q3.

Q1 is the median of first half of dataset and Q3 is the median of second half of the dataset.

(10, 20, [30], 40, 50) 60 (70, 80, [90], 100, 110)

**Q1 = 30 and Q3 = 90.**

**Inter-Quartile Range (IQR) = Q3 - Q1**

90-30 = 60

**Inter-Quartile Range (IQR) = 60**

## Variance

The**Variance**measures the average degree to which each point differs from the mean. In order to find out the variance, first calculate the difference between each point from the mean,

**square it**, and then

**average**the result.

## How to calculate the Variance?

- Step-1 : Find the mean of the dataset.
- Step-2 : Calculate difference from Mean.
- Step-3 : Square each value.
- Step-4 : Average it.

Suppose you have a dataset:

600, 470, 170, 430, 300

### Step-1 : Find the mean of the dataset

First, you have to find the**mean**of the dataset.

600 + 470 + 170 + 430 + 300
----------------------------
5

1970
-----
5

= 394

*Mean of the above dataset is*:

**394**

### Step-2 : Calculate difference from Mean

Next step is to calculate the difference of each values in the dataset from the Mean value:

Dataset : 600, 470, 170, 430, 300

600 - 394 = 206

470 - 394 = 76

170 - 394 = -224

430 - 394 = 36

300 - 394 = -94

### Step-3 : Square each value

**Square each value**.

206 * 206 = 42436

76 * 76 = 5776

(-224) * (-224) = 50176

36 * 36 = 1296

(-94) * (-94) = 8836

### Step-4 : Average it

**Average the result**.

42436 + 5776 + 50176 + 1296 + 8836
----------------------------------
5

= 108520/5

= 21704

*Variance of the above dataset is*:

**21704**

## Standard Deviation

The**Standard Deviation**is a measure of how spread out numbers are. It is calculated as the

**square root of variance**by figuring out the variation between each data point relative to the mean.

**Standard deviation (S) = square root of the variance**. Calculate the

**Standard deviation**of the following dataset:

600, 470, 170, 430, 300

*Variance of the above dataset is*:

**21704**

*Standard deviation*:

**square root of (21704) = 147**

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