Skewness and Kurtosis in Excel
This tutorial shows how to compute and interpret skewness and kurtosis in Excel using the XLSTAT software.
Dataset for computing skewness and kurtosis
The data represent the time needed to complete two online assessments, one in maths and another one in logical reasoning, by pupils of three different schools. Time is measured in minutes. Rows correspond to pupils and columns to the time spent for each of the two assessments as well as the school they belong to.
Our goal here is to study two specific characteristics of a given distribution: 1. The skewness, that reflects the asymmetry of a distribution 2. The kurtosis, that reflects the characteristics of the tails of a distribution.
Setting up the dialog box for computing skewness and kurtosis
1. Once XLSTAT is open, select the XLSTAT / Describing data / Descriptive statistics command as shown below.
2. The Descriptive Statistics dialog box appears.
3. In the General tab, select the columns corresponding to the time spent on each assessment in the Quantitative data field. Then select the column corresponding to the school name in the Subsamples field. We also want to display Variable-Category labels in the output. These include the variable name as a prefix and the category name as a suffix. Finally, select the Sheet option in order to display the results on a new sheet and the Sample labels to consider the first row of the data table as labels.
4. In the Options tab, activate the following options.
- In the Outputs tab, click on the All button to select all the statistics for quantitative data. You can also select the statistics you’re interested in one by one.
How to interpret Skewness and Kurtosis
The results are displayed on a new sheet named Desc. A full set of descriptive statistics is displayed per school (columns C-H). XLSTAT proposes several coefficients of skewness and kurtosis. In this example, we will be referring to the Fisher coefficients which are not biased on the assumption that the data is normally distributed.
Formulas can be found in the XLSTAT Help menu (click on the Help button in the dialog box).
a. Interpreting the skewness
Skewness measures the asymmetry of a distribution. A distribution is called asymmetric when one tail is longer than the other. If the skewness is positive, then the distribution is skewed to the right while a negative skewness implies a distribution skewed to the left. A zero skewness suggests a perfectly symmetric distribution.
In this part, we will interpret results related to the maths assessment (see below).
The three samples seem to have contrasted skewness coefficients:
- Sample A has a strong positive skewness (1.42). This reflects a long distribution tail on the right.
- Sample B has a strong negative skewness (-1.63). This reflects a long distribution tail on the left
- A zero skewness is estimated for sample C. In fact, the median of sample C (49.8) is almost identical to the mean value (49.6).
Histograms allow us to confirm the above observations. The top histogram (sample A) shows a distribution skewed to the right, the second one (sample B) is a distribution skewed to the left while the third one (sample C) a symmetric one.
b. Interpreting the kurtosis
Kurtosis provides information on the tails (the extremes, or outliers) of a distribution. When interpreting kurtosis, the normal distribution is used a reference. A positive kurtosis implies a distribution with more extreme possible data values (outliers) than a normal distribution thus fatter tails (Leptokurtic distributions). A negative kurtosis implies a distribution with less extreme possible data values than a normal distribution thus thinner tails (Platykurtic distributions). Finally, distributions with zero kurtosis have roughly the same outlier character as a normal distribution (Mesokurtic distributions).
In this section, we will interpret the results related to the second assessment (see below). Based on the coefficients above, the shape of the three distributions differ in terms of kurtosis:
- A positive kurtosis is estimated for School A (5.40)
- A negative one is estimated for School B (-1.32)
- A zero kurtosis was detected for School C.
Histograms confirm these observations. The top histogram (sample A) shows a leptokurtic distribution, while the second one (sample B) shows a platykurtic distribution. The third one (sample C) displays a distribution with a shape similar to the shape of a normal distribution.
What’s next: How to visualize skewness and kurtosis
The Histogram is commonly used to examine the distribution of numerical data. It allows to observe the tail and the peak of a frequency distribution in a single chart. On the top, it provides information on the central tendency, data dispersion as well as the presence of outliers. Here’s how to generate histograms in XLSTAT.
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