# Measuring Data Spread: Explained with Standard Deviation and Variance

## What are Standard Deviation and Variance?

Standard deviation and variance are the fundamental types of statistical measures that are useful for computing the spread of data or calculating the dispersion of data observations within a data set. Data variability can be determined by the deviation of these statistical measures from the expected value, which indicates how different the data observations are from each other. Let’s discuss both statistical measures.### Standard Deviation

In statistics, the measure of dispersion with a dataset that shows the average amount by which each data observation in the given set differs from the expected value is said to be the standard deviation. This is the square root of the variance. The basic moto of this statistical measure is to find how the data observations are spread out from the expected value. When the standard deviation is smaller, it indicates that the data observations are closely clustered around the expected, while when it is larger, it indicates that the data observations are more widely dispersed. Below steps will help let you know how to evaluate the standard deviation.### Variance

## Formulas of Standard Deviation and Variance

## How Standard Deviation and Variance Are Helpful to Measure the Spread of Data?

### Quantifying Variability

### Comparing Datasets

### Outlier Detection

### Decision Making

## Examples of Standard Deviation and Variance

Here are a few examples of Standard Deviation and Variance.### Example of Standard Deviation

What is the standard deviation of the monthly sales data of a company for the past year, given the sales values [100, 120, 80, 140, 90, 135, 95, 110, 115, 85, 120, 75]?Solution

Step 1: Find the expected value of the yearly sales of a company.

Yearly sales = 100, 120, 80, 140, 90, 135, 95, 110, 115, 85, 120, 75

Expected value = [100 + 120 + 80 + 140 + 90 + 135 + 95 + 110 + 115 + 85 + 120 + 75] / 12

Expected value = 1265/12

Expected value = 105.42

Step 2: Calculate the deviation of each monthly sale from the expected value

Deviation = [100 – 105.45, 120 – 105.45, 80 – 105.45, 140 – 105.45, 90 – 105.45, 135 – 105.45, 95 – 105.45, 110 – 105.45, 115 – 105.45, 85 – 105.45, 120 – 105.45, 75 – 105.45]

Deviation = [-5.42, 14.58, -25.42, 34.58, -15.42, 29.58, -10.42, 4.58, 9.58, -20.42, 14.58, -30.42]

Step 3: Now take the square of each deviation.

Square of Deviation = [(-5.42)2, (14.58)2, (-25.42)2, (34.58)2, (-15.42)2, (29.58)2, (-10.42)2, (4.58)2, (9.58)2, (-20.42)2, (14.58)2, (-30.42)2]

Squared Deviation = [29.38, 212.58, 646.18, 1195.78, 237.78, 874.98, 108.58, 20.98, 91.78, 416.98, 212.58, 925.38]

Step 4: Now take the average of the squared deviations.

Sum of Squared Deviation = [29.38 + 212.58 + 646.18 + 1195.78 + 237.78 + 874.98 + 108.58 + 20.98 + 91.78 + 416.98 + 212.58 + 925.38]

Sum of Squared Deviation = 4972.96

Average of Squared Deviation = 4972.96/12

Average of Squared Deviation = 414.41

Step 5: To evaluate the standard deviation, take the square root of the average squared deviation.

standard deviation = √414.41

standard deviation = 20.36

### Example of Variance

Solution

Step 1: Find the expected value of the city’s temperature in two weeks.

City’s temperature = 34, 35, 36, 33, 32, 34, 38, 37, 35, 33, 34, 39, 40, 41

Expected value = [34 + 35 + 36 + 33 + 32 + 34 + 38 + 37 + 35 + 33 + 34 + 39 + 40 + 41] / 14

Expected value = 501/14

Expected value = 35.79

Step 2: Calculate the deviation of each day's temperature from the expected value

Deviation = [34 – 35.79, 35 – 35.79, 36 – 35.79, 33 – 35.79, 32 – 35.79, 34 – 35.79, 38 – 35.79, 37 – 35.79, 35 – 35.79, 33 – 35.79, 34 – 35.79, 39 – 35.79, 40 – 35.79, 41 – 35.79]

Deviation = [-1.79, -0.79, 0.21, -2.79, -3.79, -1.79, 2.21, 1.21, -0.79, -2.79, -1.79, 3.21, 4.21, 5.21]

Step 3: Now take the square of each deviation.

Square of Deviation = [(-1.79)2, (-0.79)2, (0.21)2, (-2.79)2, (-3.79)2, (-1.79)2, (2.21)2, (1.21)2, (-0.79)2, (-2.79)2, (-1.79)2, (3.21)2, (4.21)2, (5.21)2]

Squared Deviation = [3.20, 0.62, 0.04, 7.78, 14.36, 3.20, 4.88, 1.46, 0.62, 7.78, 3.20, 10.30, 17.72, 27.14]

Step 4: Now take the average of the squared deviations.

Sum of Squared Deviation = [3.20 + 0.62 + 0.04 + 7.78 + 14.36 + 3.20 + 4.88 + 1.46 + 0.62 + 7.78 + 3.20 + 10.30 + 17.72 + 27.14]

Sum of Squared Deviation = 102.3

Average of Squared Deviation = 102.3/14

Average of Squared Deviation = 7.311