# Descriptive statistics for stock returns

Descriptive statistics offer a simple way of understanding distributions of stock returns. They give us an idea about a distribution’s centrality, dispersion, and other features. In this tutorial, we will show you how to generate descriptive statistics for stock returns using Excel’s data analysis tool.

## Using Excel to get descriptive statistics for stock returns

What you need to proceed with is an Excel spreadsheet with the returns of a stock (or another asset) recorded in a single column. Then, follow the steps below.

1. If you haven’t used the data analysis tool in Excel before, make sure to enable the “Analysis ToolPak.”
2. Then, go to the “Data” tab and click on “Data Analysis.”
3. In the menu that shows up, choose “Descriptive statistics” and hit the OK button.
4. For the input range choose the column where the returns are recorded.
5. If the first row is a label (e.g., stock ticker), tick the Labels in First Row box.
6. Choose your output range (this is where the results will be shown).
7. Tick the summary statistics box and click the OK button.

That’s it! You should see a table similar to Table 1 below.

## Interpreting mean return and median return

The statistics shown in Table 1 are based on the monthly returns of a stock over 30 years. So, there are 30 * 12 = 360 return observations in total as shown by the count. The average return is given by the mean and is equal to 0.020728949, or 2.07%. The median return is 2.21%, which means there are 360 / 2 = 180 months with monthly returns of less than 2.21% and 180 months with returns of more than 2.21%. Both the mean and the median give us an idea about the centrality of stock returns. In other words, they inform us about the average performance of the stock over the sample period.

## Understanding standard deviation of returns and variance of returns

First, let’s explain the relation between standard deviation and variance: Standard deviation is simply the square root of variance. Based on the data in Table 1, the variance is 0.007964134. Taking the square root yields √(0.007964134) = 0.089241997, which is indeed the standard deviation figure in Table 1.

Both standard deviation and variance capture how dispersed the return observations are around their mean. Specifically, a tight distribution around the mean would have low standard deviation and variance. Conversely, if there were many returns observations far from the average, then the standard deviation and variance would both be high.

When computing the variance, we take the square of return observations, which makes it harder to interpret vis-à-vis the mean return. Since standard deviation is the square root of the variance, it can be more readily compared to the mean return. For example, based on the data in Table 1, we can say that the stock has an average return of 2.07% with a standard deviation of 8.92%. Therefore, standard deviation captures return volatility and is a measure of risk.

## Skewness, kurtosis, and tails of a distribution

Skewness is about the symmetry of a distribution. For example, the normal distribution is symmetric around its mean. Symmetric distributions have a skewness of zero. It is common for stock return distributions to be asymmetric. A positive skewness implies a long tail to the right. That is, there would be cases of highly positive return observations far away from the average return. Conversely, a negative skewness suggests a long tail to the left. In this case, highly negative returns would be observed below the average return.

Kurtosis is all about the heaviness of the tails of a distribution. The normal distribution has a kurtosis of 3. For example, if a distribution has a kurtosis of 5, it is said to have an excess kurtosis of 5 − 3 = 2. Distributions that have positive excess kurtosis are considered to have fat tails (also known as leptokurtic distributions). This implies that extreme observations are more common in such distributions relative to otherwise similar normal distributions. Stock return distributions tend to be fat-tailed. Therefore, investors should be wary of the fact that they are likely to experience extreme return observations more often than they might like to believe.

## Video tutorial: Descriptive statistics for stock returns

You can also watch here our Youtube video titled “Descriptive statistics for stock returns | Analyzing stock returns #4.” It offers a step-by-step guide to getting descriptive statistics for stock returns using Excel.

Descriptive statistics for stock re...
Descriptive statistics for stock returns | Analyzing stock returns #4

##### Recap

In this tutorial, we illustrate how to use Excel’s “Descriptive statistics” analysis tool to obtain descriptive statistics of stock returns. Furthermore, we discuss mean return and median return as measures of centrality. Return volatility is captured by standard deviations of returns. Finally, skewness is used to examine the symmetry of returns, and kurtosis is a useful statistic to analyze whether stock return distributions have fat tails.

##### What is next?

This post is part of the series on analyzing stock returns. In the previous tutorial, we demonstrated how to plot a histogram of stock returns using Excel.

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