Variance is a measure of how spread out a set of data is. It is a statistical concept that is used to describe the dispersion of a set of data around its mean. Variance is an important tool in statistics because it allows researchers to understand the distribution of a set of data and make predictions about future data points.

There are two types of variance: population variance and sample variance. Population variance is a measure of the dispersion of a set of data in a population. It is calculated by taking the sum of the squared differences between each value in the population and the population mean, and dividing by the number of values in the population.

Sample variance is a measure of the dispersion of a set of data in a sample. It is calculated in a similar way to population variance, but the sample size is used instead of the population size.

### The formula for calculating population variance is:

Population variance = (sum of squared differences between each value in the population and the population mean) / (number of values in the population)

The formula for calculating sample variance is:

Sample variance = (sum of squared differences between each value in the sample and the sample mean) / (sample size – 1)

The reason that the sample size is subtracted by one is because the sample variance is an estimate of the population variance. As the sample size increases, the sample variance becomes a more accurate estimate of the population variance.

One of the key properties of variance is that it is always positive. This means that if the data is spread out, the variance will be high, and if the data is concentrated around the mean, the variance will be low. Variance is also a measure of how far the data points are from the mean. If the data points are far from the mean, the variance will be high, and if the data points are close to the mean, the variance will be low.

Variance is often used in conjunction with other statistical measures, such as standard deviation. Standard deviation is a measure of the dispersion of a set of data that is calculated by taking the square root of the variance. Standard deviation is often used because it is easier to interpret than variance, as it is expressed in the same units as the data.

There are several ways that variance can be used in statistics. One common use is to compare the variance of two or more sets of data. For example, a researcher might compare the variance of the heights of men and women to see if there is a significant difference between the two groups.

Variance is also used in hypothesis testing. A hypothesis is a statement about a population that is being tested. For example, a researcher might have a hypothesis that the average income for a certain group of people is higher than the average income for the general population. To test this hypothesis, the researcher would collect a sample of data from the group of people and use variance to see if the sample variance is significantly different from the population variance.

In addition to its use in hypothesis testing, variance is also used in statistical models. Statistical models are used to make predictions about future data based on past data. Variance is often used in statistical models to help understand the uncertainty of the predictions.

To calculate the variance of a set of data, you need to follow these steps:

- Calculate the mean of the data. To do this, add up all the values in the data set and divide by the number of values.
- Calculate the squared difference between each value in the data set and the mean. To do this, subtract the mean from each value and then square the result.
- Add up the squared differences for each value in the data set.
- Divide the sum of the squared differences by the number of values in the population (for population variance) or the sample size minus one (for sample variance). This will give you the variance of the data set.

It is important to note that the variance is always a positive value. This means that if the data is spread out, the variance will be high, and if the data is concentrated around the mean, the variance will be low. Variance is also a measure of how far the data points are from the mean. If the data points are far from the mean, the variance will be high, and if the data points are close to the mean, the variance will be low.

**Suppose a researcher is studying the heights of a sample of 50 adults. The heights of the adults are as follows:**

68 inches, 70 inches, 69 inches, 72 inches, 71 inches, 67 inches, 65 inches, 66 inches, 73 inches, 72 inches, 70 inches, 71 inches, 68 inches, 73 inches, 72 inches, 71 inches, 69 inches, 68 inches, 71 inches, 70 inches, 69 inches, 72 inches, 68 inches, 72 inches, 70 inches, 71 inches, 73 inches, 71 inches, 72 inches, 68 inches, 69 inches, 71 inches, 68 inches, 71 inches, 70 inches, 68 inches, 70 inches, 73 inches, 72 inches, 71 inches, 68 inches, 70 inches, 73 inches, 71 inches, 68 inches, 72 inches, 70 inches, 69 inches, 71 inches

To calculate the variance of these data, follow these steps:

- Calculate the mean height of the sample by adding up all the heights and dividing by the number of people in the sample. In this case, the mean height is 70 inches.
- Calculate the squared difference between each height and the mean height. To do this, subtract the mean height from each height and then square the result. The squared differences are as follows:

4 inches, 0 inches, 1 inches, 4 inches, 1 inches, 3 inches, 5 inches, 4 inches, 1 inches, 2 inches, 0 inches, 1 inches, 2 inches, 1 inches, 2 inches, 1 inches, 1 inches, 2 inches, 0 inches, 1 inches, 1 inches, 2 inches, 2 inches, 0 inches, 2 inches, 1 inches, 1 inches, 1 inches, 2 inches, 4 inches, 1 inches, 1 inches, 2 inches, 0 inches, 1 inches, 4 inches, 1 inches, 2 inches, 0 inches, 1 inches, 4 inches, 1 inches, 2 inches, 4 inches, 1 inches, 2 inches, 0 inches, 1 inches

- Add up the squared differences. In this case, the sum of the squared differences is 64 inches.
- Divide the sum of the squared differences by the sample size minus one. In this case, the sample variance is (64 inches) / (50 – 1) = 1.28 inches.

This means that the heights of the adults in the sample are spread out by an average of 1.28 inches. The researcher can use this information to make predictions about the heights of adults in the population. For example, if the mean height of the sample is 70 inches, the researcher might predict that the heights of adults in the population are likely to fall within a range of 68.72 inches to 71.28 inches (since the variance is 1.28 inches).

In conclusion, variance is a measure of how spread out a set of data is. It is an important tool in statistics because it allows researchers to understand the distribution of a set of data and make predictions about future data points. It is often used in conjunction with other statistical measures, such as standard deviation, and it is used in hypothesis testing and statistical modeling.