Sample Mean & Variance: Definition, Equations & Examples

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Sample mean and variance are both important statistics that can you can use to make predictions about a population. In this lesson, learn how to calculate these important values.

How Tall Are Those Bushes?

Tony owns a plant nursery, and one of his biggest sellers is blueberry bushes. He sells the bushes to his customers when they are at least 18 inches tall. Tony wants to know how long it will take each of his blueberry bushes to grow tall enough to sell. To get an estimate of this time, he selects ten plants at random and records the number of days each one takes to grow from a seed into an 18 inch tall plant.

Time each plant required to grow to 18 inches tall
plant height data

Sample Mean

A sample is a set of measurements taken from a larger population. In this case, the population would be all of Tony's blueberry bushes, and the sample would just include the specific ten bushes he selected to observe. Tony's measurements represent a random sample because they were selected at random from the population. Each seed had an equal chance of being chosen for the sample. In order for a sample to give a good approximation of the population, it must be randomly selected.

The sample mean is simply the average of all the measurements in the sample. If the sample is random, then the sample mean can be used to estimate the population mean.

Sample mean equation:


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For Tony's data, the sample mean is:


sample mean example


Sample Variance

Another important statistic that can be calculated for a sample is the sample variance. Variance measures how spread out the data in a sample is. Two samples can have the same mean, but be distributed very differently. Variance is one way to quantify these differences. The variance of a sample is also closely related to the standard deviation, which is simply the square root of the variance. The symbol typically used to represent standard deviation is s, so the symbol for variance is s2.

To find the sample variance, follow these steps:

  • First, calculate the sample mean.
  • Next, subtract the mean value from the value of each measurement.
  • Square the resulting values.
  • Add the results together to get the sum of squared deviations from the mean.
  • Finally, divide this by the number of degrees of freedom, which is equal to the total number of measurements minus one (n -1)

In equation form, this looks like:


variance equation


The easiest way to do this is to make a table like this:


variance table


For Tony's data, the sample variance is equal to 43.344.

Standard deviation often gives you more useful information than variance. About 70% of the values in the population are expected to fall within one standard deviation on each side of the mean. To find the standard deviation from the variance, simply take the square root.


standard deviation example


Since the mean number of days in Tony's sample was 96.7, he can expect about 70% of his trees to reach 18 inches tall between 90 days and 103 days.

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