Variance is a measure of the spread or variability of a distribution. It measures how far the values in a dataset are from the mean of the distribution. Here's how to calculate the variance step by step using an example:

Consider the following dataset of exam scores: 85, 90, 75, 80, 95.

**Step 1**: **Calculate the mean**

Add up all the values in the dataset and divide by the total number of values to find the mean:

```
85 + 90 + 75 + 80 + 95 / 5 = 85
The mean is 85.
```

**Step 2**: **Calculate the deviations from the mean**

Calculate the deviation of each value from the mean by subtracting the mean from each value:

```
85 - 85 = 0
90 - 85 = 5
75 - 85 = -10
80 - 85 = -5
95 - 85 = 10
```

**Step 3**: **Square the deviations**

Square each deviation to eliminate the negative values:

```
0^2 = 0
5^2 = 25
(-10)^2 = 100
(-5)^2 = 25
10^2 = 100
```

**Step 4**: **Calculate the sum of the squared deviations**

Add up all the squared deviations:

```
0 + 25 + 100 + 25 + 100 = 250
```

**Step 5**: **Calculate the variance**

Divide the sum of the squared deviations by the total number of values in the dataset:

```
250 / 5 = 50
The variance is 50.
```

**Step 6**: **Calculate the standard deviation**

The standard deviation is the square root of the variance. In this example, the standard deviation is:

```
√50 = 7.07
The standard deviation is 7.07
Therefore, the variance of the exam scores is 50,
and the standard deviation is 7.07.
```

This means that the exam scores are somewhat spread out from the mean of 85. A higher variance and standard deviation indicate that the data is more spread out, while a lower variance and standard deviation indicate that the data is more tightly clustered around the mean.

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