We're here to help you navigate STEM learning at home while schools and camps are closed due to COVID-19.

Here are some resources to guide your at home learning:

Create Assignment

This feature requires that you be logged in as a Google Classroom teacher and that you have an active class in Google Classroom.

If you are a Google Classroom teacher, please log in now.

For additional information about using Science Buddies with Google Classroom, see our FAQ.

It would be useful to have a measure of scatter that has the following properties:

  1. The measure should be proportional to the scatter of the data (small when the data are clustered together, and large when the data are widely scattered).
  2. The measure should be independent of the number of values in the data set (otherwise, simply by taking more measurements the value would increase even if the scatter of the measurements was not increasing).
  3. The measure should be independent of the mean (since now we are only interested in the spread of the data, not its central tendency).

Both the variance and the standard deviation meet these three criteria for normally-distributed (symmetric, "bell-curve") data sets.

The variance (σ2) is a measure of how far each value in the data set is from the mean. Here is how it is defined:

  1. Subtract the mean from each value in the data. This gives you a measure of the distance of each value from the mean.
  2. Square each of these distances (so that they are all positive values), and add all of the squares together.
  3. Divide the sum of the squares by the number of values in the data set.

The standard deviation (σ) is simply the (positive) square root of the variance.

The Summation Operator

In order to write the equation that defines the variance, it is simplest to use the summation operator, Σ. The summation operator is just a shorthand way to write, "Take the sum of a set of numbers." As an example, we'll show how we would use the summation operator to write the equation for calculating the mean value of data set 1. We'll start by assigning each number to variable, X1X6, like this:

Data Set 1
Variable Value
X1 3
X2 4
X3 4
X4 5
X5 6
X6 8

Think of the variable (X) as the measured quantity from your experiment—like number of leaves per plant—and think of the subscript as indicating the trial number (1–6). To calculate the average number of leaves per plant, we first have to add up the values from each of the six trials. Using the summation operator, we'd write it like this:

Equation for the summation of six different values of X using the sigma symbol

which is equivalent to:

Equation adds six different values of X together

or:

Equation adding three, four, four, five, six and eight to equal thirty

Obviously the sum is a lot more compact to write with the summation operator. Here is the equation for calculating the mean, μx, of our data set using the summation operator:

Equation for the average value of X is equal to the summation of all X values divided by six

The general equation for calculating the mean, μ, of a set of numbers, X1XN, would be written like this:

Equation for the average value of X is equal to the summation of all X values divided by the number of values N

Sometimes, for simplicity, the subscripts are left out, as we did on the right, above. Doing away with the subscripts makes the equations less cluttered, but it is still understood that you are adding up all the values of X.

The Equation Defining Variance

Now that you know how the summation operator works, you can understand the equation that defines the population variance (see note at the end of this page about the difference between population variance and sample variance, and which one you should use for your science project):

Equation for variance equals the sum of X values minus the average of X values squared divided by the number of values N

The variance (σ2), is defined as the sum of the squared distances of each term in the distribution from the mean (μ), divided by the number of terms in the distribution (N).

There's a more efficient way to calculate the standard deviation for a group of numbers, shown in the following equation:

Equation for variance equals the sum of X values squared divided by the number of values N minus the mean of X values squared

You take the sum of the squares of the terms in the distribution, and divide by the number of terms in the distribution (N). From this, you subtract the square of the mean (μ2). It's a lot less work to calculate the standard deviation this way.

It's easy to prove to yourself that the two equations are equivalent. Start with the definition for the variance (Equation 1, below). Expand the expression for squaring the distance of a term from the mean (Equation 2, below).

Five steps show how the equation for variance is simplified by multiplying out exponents

Now separate the individual terms of the equation (the summation operator distributes over the terms in parentheses, see Equation 3, above). In the final term, the sum of μ2/N, taken N times, is just 2/N.

Next, we can simplify the second and third terms in Equation 3. In the second term, you can see that ΣX/N is just another way of writing μ, the average of the terms. So the second term simplifies to −2μ2 (compare Equations 3 and 4, above). In the third term, N/N is equal to 1, so the third term simplifies to μ2 (compare Equations 3 and 4, above).

Finally, from Equation 4, you can see that the second and third terms can be combined, giving us the result we were trying to prove in Equation 5.

As an example, let's go back to the two distributions we started our discussion with:

data set 1: 3, 4, 4, 5, 6, 8
data set 2: 1, 2, 4, 5, 7, 11 .

What are the variance and standard deviation of each data set?

We'll construct a table to calculate the values. You can use a similar table to find the variance and standard deviation for results from your experiments.

Data Set N ΣX ΣX2 μ μ2 σ2 σ
1 6 30 166 5 25 2.67 1.63
2 6 30 216 5 25 11.00 3.32

Although both data sets have the same mean (μ = 5), the variance (σ2) of the second data set, 11.00, is a little more than four times the variance of the first data set, 2.67. The standard deviation (σ) is the square root of the variance, so the standard deviation of the second data set, 3.32, is just over two times the standard deviation of the first data set, 1.63.

Example frequency histogram of plants with a certain number of leaves and low dispersion

A histogram showing the number of plants that have a certain number of leaves. All plants have a different number of leaves ranging from 3 to 8 (except for 2 plants that have 4 leaves). The difference between the highest number of leaves and lowest number of leaves is 5 so the data has relative low variance.



Example frequency histogram of plants with a certain number of leaves and high dispersion

A histogram showing the number of plants that have a certain number of leaves. All plants have different number of leaves ranging from 1 to 11. The difference between the plant with the highest number of leaves and the lowest number of leaves is 10, so the data has relatively high variance.

The variance and the standard deviation give us a numerical measure of the scatter of a data set. These measures are useful for making comparisons between data sets that go beyond simple visual impressions.

Population Variance vs. Sample Variance

The equations given above show you how to calculate variance for an entire population. However, when doing science project, you will almost never have access to data for an entire population. For example, you might be able to measure the height of everyone in your classroom, but you cannot measure the height of everyone on Earth. If you are launching a ping-pong ball with a catapult and measuring the distance it travels, in theory you could launch the ball infinitely many times. In either case, your data is only a sample of the entire population. This means you must use a slightly different formula to calculate variance, with an N-1 term in the denominator instead of N:

Equation for variance in a sample population using the population N minus one

This is known as Bessel's correction.

Free science fair projects.