Measures of Central Tendency in Descriptive Statistics

Chapter 2: Descriptive Statistics

Chapter Outline

• Frequency Distributions and Their Graphs

• More Graphs and Displays

• Measures of Central Tendency

• Measures of Variation

• Measures of Position

Measures of Central Tendency

Introduction

Measures of central tendency are statistical values that represent a typical, or central, entry in a data set. They are essential for summarizing and describing data distributions in statistics. The three most common measures are the mean, median, and mode.

Mean

The mean, often called the average, is the sum of all data entries divided by the number of entries. It is a widely used measure because it incorporates every value in the data set.

• Population Mean ($\mu$): The mean of all members in a population.

• Sample Mean ($\bar{x}$): The mean of a sample drawn from the population.

Formulas:

• Population mean: $\mu = \frac{\Sigma x}{N}$

• Sample mean: $\bar{x} = \frac{\Sigma x}{n}$

Example: Given the weights (in pounds) for a sample of adults: 274, 235, 223, 268, 290, 285, 235. Sum: $274 + 235 + 223 + 268 + 290 + 285 + 235 = 1810$ Mean: $\bar{x} = \frac{1810}{7} \approx 258.6$ pounds.

Median

The median is the value that lies in the middle of an ordered data set. It divides the data into two equal parts and is less affected by outliers than the mean.

• If the number of entries is odd, the median is the middle entry.

• If the number of entries is even, the median is the mean of the two middle entries.

Example: For the ordered weights: 223, 235, 235, 268, 274, 285, 290 (7 entries, odd), the median is the 4th entry: 268 pounds. If one entry (285) is removed, the ordered set is: 223, 235, 235, 268, 274, 290 (6 entries, even). Median is $\frac{235 + 268}{2} = 251.5$ pounds.

Mode

The mode is the data entry that occurs with the greatest frequency. A data set may have no mode, one mode (unimodal), or more than one mode (bimodal, multimodal).

• If no entry is repeated, there is no mode.

• If two or more entries occur with the same greatest frequency, each is a mode.

Example: For the weights: 223, 235, 235, 268, 274, 285, 290, the mode is 235 (occurs twice).

Example (Categorical Data): In a survey, the most frequent response for political party affiliation is 'Democrat', so the mode is 'Democrat'.

Comparing Mean, Median, and Mode

All three measures describe a typical entry of a data set, but each has advantages and disadvantages.

• Mean: Uses all data entries, but is sensitive to outliers.

• Median: Not affected by outliers, represents the center of the data.

• Mode: May not always represent a typical value, especially if the data is spread out.

Example: Ages in a class: 20, 20, 20, 21, 21, 22, 22, 23, 23, 23, 24, 24, 65. Mean: $\bar{x} \approx 23.8$ years Median: 21.5 years Mode: 20 years The mean is influenced by the outlier (65), while the median is not. The mode may not represent a typical age.

Weighted Mean

The weighted mean is used when data entries have different weights or importance. It is calculated as:

$\bar{x} = \frac{\Sigma xw}{\Sigma w}$

• $x$ = data value

• $w$ = weight of each value

Example: Calculating grade point average (GPA) using grades and credit hours.

Mean of Grouped Data

When data is grouped into classes, the mean can be estimated using class midpoints and frequencies:

$\bar{x} = \frac{\Sigma (x f)}{n}$

• $x$ = midpoint of each class

• $f$ = frequency of each class

• $n$ = total number of data entries

Steps:

1. Find the midpoint of each class: $\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}$

2. Multiply each midpoint by its class frequency.

3. Sum the products and divide by the total frequency.

Shapes of Distributions

The shape of a data distribution affects the relationship between mean and median.

• Symmetric Distribution: The left and right halves are mirror images. Mean and median are approximately equal.

• Uniform Distribution: All classes have equal or nearly equal frequencies. Also symmetric.

• Skewed Left (Negatively Skewed): The tail extends to the left. Mean is less than median.

• Skewed Right (Positively Skewed): The tail extends to the right. Mean is greater than median.

Distribution Shape	Mean vs. Median	Description
Symmetric	Mean ≈ Median	Halves are mirror images
Uniform	Mean ≈ Median	All frequencies equal
Skewed Left	Mean < Median	Tail to the left
Skewed Right	Mean > Median	Tail to the right