Describing Data Numerically: Measures of Central Tendency and Dispersion

Describing Data Numerically

Measures of Central Tendency

Measures of central tendency summarize a set of data by identifying the central position within that set of data. The most common measures are mean, median, and mode.

• Mean: The arithmetic average of a data set. Add up all data values and divide by the number of values. Formula: $\bar{x} = \frac{\sum x_i}{n}$ Excel Formula: =AVERAGE(range) Best Used: When the data is roughly symmetric and has no extreme values (outliers).

• Median: The middle value when data is sorted. If the number of values is even, the median is the average of the two middle values. Excel Formula: =MEDIAN(range) Best Used: When the data has outliers or is skewed, as the median is resistant to outliers. Also preferred for open-ended distributions (e.g., "50 and older").

• Mode: The value that occurs most frequently in a data set. There can be more than one mode (bimodal, multimodal) or no mode if all values are unique. Excel Formula: =MODE(range)

• Midrange: The average of the minimum and maximum values in the data set. Formula: $\text{Midrange} = \frac{\text{MAX} + \text{MIN}}{2}$ Excel Formula: =(MAX(range) + MIN(range))/2

• Weighted Mean: Used when data values have different weights. Multiply each value by its weight, sum the results, then divide by the sum of the weights. Formula: $\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$

Measures of Dispersion

Measures of dispersion describe the spread or variability of the data. Common measures include range, standard deviation, and variance.

• Range: The difference between the maximum and minimum values in the data set. Formula: $\text{Range} = \text{MAX} - \text{MIN}$ Excel Formula: =MAX(range) - MIN(range)

• Standard Deviation (SD): Measures the average distance of data values from the mean. A higher standard deviation indicates more spread out data. Formula: $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$ Excel Formula: =STDEV.S(range)

• Variance: The square of the standard deviation. It represents the average squared deviation from the mean. Formula: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$ Excel Formula: =VAR.S(range)

Parameters vs. Statistics

It is important to distinguish between parameters and statistics:

• Parameters: Measures calculated from entire populations.

• Statistics: Measures calculated using sample data sets.

Summary Table: Measures of Central Tendency and Dispersion

Measure	Definition	Best Used When	Excel Formula
Mean	Arithmetic average	Symmetric distribution, no outliers	=AVERAGE(range)
Median	Middle value	Skewed data, outliers, open-ended	=MEDIAN(range)
Mode	Most frequent value	Nominal data, categorical data	=MODE(range)
Midrange	Average of min and max	Quick estimate of center	=(MAX(range)+MIN(range))/2
Weighted Mean	Mean with weights	Values have different importance	Additional info: Use SUMPRODUCT(range,weights)/SUM(weights)
Range	Max - Min	Simple measure of spread	=MAX(range)-MIN(range)
Standard Deviation	Average distance from mean	Assessing variability	=STDEV.S(range)
Variance	Square of SD	Statistical calculations	=VAR.S(range)

Examples

• Mean Example: For the data set {2, 4, 6, 8}, the mean is $(2+4+6+8)/4 = 5$.

• Median Example: For the data set {2, 4, 6, 8}, the median is $(4+6)/2 = 5$.

• Mode Example: For the data set {2, 4, 4, 6}, the mode is 4.

• Range Example: For the data set {2, 4, 6, 8}, the range is $8-2=6$.

• Standard Deviation Example: For the data set {2, 4, 6, 8}, calculate the mean (5), then SD: $\sqrt{\frac{(2-5)^2+(4-5)^2+(6-5)^2+(8-5)^2}{4-1}} = \sqrt{\frac{9+1+1+9}{3}} = \sqrt{\frac{20}{3}} \approx 2.58$.