16.11: Microsoft Excel: Finding Central Tendency, Skew, and Kurtosis

Central tendency refers to the central point or typical value of a dataset. It summarizes the data set with a single value that represents the center of its distribution. The three main measures of central tendency are:

Mean: The arithmetic average of all data points. It is calculated by adding all the values together and dividing by the number of values. The mean is sensitive to extreme values (outliers).

Median: The middle value when the data points are arranged in ascending or descending order. If there is an even number of observations, the median is the average of the two middle numbers. The median is less affected by outliers and skewed data.

Mode: The most frequently occurring value in a dataset. A dataset may have one mode, more than one mode, or no mode at all.

Variation measures the spread or dispersion of a set of data points. It provides insights into how much the data points differ from the mean and from each other. Key measures of variation include:

Range: The difference between the maximum and minimum values in the dataset. It provides a quick sense of the spread but is highly sensitive to outliers.

Variance: The average of the squared differences from the mean. It quantifies how spread out the data points are around the mean.

Standard Deviation: The square root of the variance. It is expressed in the same units as the data and provides a measure of the average distance of each data point from the mean.

Skew

Skewness measures the asymmetry of the data distribution around the mean. It indicates whether the data points are more concentrated on one side of the distribution or the side to which the tail is longer or fatter. Types of skewness include:

Positive Skew (Right Skew): The right tail is longer or fatter than the left. The mean is greater than the median.

Negative Skew (Left Skew): The left tail is longer or fatter than the right. The mean is less than the median.

A skewness value close to zero indicates that the data distribution is symmetric.

Kurtosis

Kurtosis measures the "tailedness" or the sharpness of the peak of a data distribution. It provides insight into the extremities (tails) of the distribution. Types of kurtosis include:

Positive Kurtosis (Leptokurtic): Indicates a distribution with a sharper peak and heavier tails than a normal distribution. Data points are more concentrated in the tails and the peak.

Negative Kurtosis (Platykurtic): Indicates a distribution with a flatter peak and lighter tails than a normal distribution. Data points are less concentrated in the tails and the peak.

Mesokurtic: Indicates a distribution with kurtosis similar to that of a normal distribution.

Kurtosis helps in understanding the outliers and the probability of extreme values in the data set.

Consider the following three hypothetical datasets plotted in Microsoft Excel.

Now, three measures of the central tendency — mean, median, and mode — can be calculated using the Excel functions AVERAGE, MEDIAN, and MODE.SNGL for the selected range of data.

After visualizing datasets A and C, their tails appear extended to the left and right.

Quantitatively, this skewness can be determined using the Excel function SKEW for the selected range of data. They are calculated as follows for datasets A, B, and C.

Kurtosis can be measured using the Excel function KURT for the selected range of data for the datasets A, B, and C.

Skewness and kurtosis, as measures of asymmetry, determine the departure from symmetry. Skew values closer to zero indicate more symmetry in the data.

Positive kurtosis values indicate a higher peak of the distribution compared to a normal distribution, with fewer values in the center than in the tails.

Conversely, negative values show the distribution's flatness, with more values in the center than in the tails.