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 the main features of a collection of data. The three most common measures are the mean, median, and mode.

Mean

The mean, often referred to as the average, is calculated by dividing the sum of all data entries 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 subset (sample) of the population.

Formulas:

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

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

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 fourth entry: 268 pounds. If one entry (285) is removed (6 entries, even): 223, 235, 235, 268, 274, 290. Median is the mean of the third and fourth entries: $\frac{235 + 268}{2} = 251.5$ pounds.

Mode

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

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

• If two entries occur with the same greatest frequency, the set is bimodal.

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

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; 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 varying weights. It is calculated by multiplying each entry by its weight, summing these products, and dividing by the sum of the weights.

Formula:

• $\bar{x} = \frac{\sum xw}{\sum w}$

Example: If grades and credit hours are given, assign points (A=4, B=3, C=2, D=1, F=0) and calculate the weighted mean (grade point average).

Mean of Grouped Data

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

Formula:

• $\bar{x} = \frac{\sum xf}{n}$

Where $x$ is the class midpoint and $f$ is the frequency of the class.

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 distribution provides insight into the nature of the data and the relationship between mean and median.

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

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

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

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

Distribution Type	Shape	Mean vs. Median
Symmetric	Mirror image halves	Mean = Median
Uniform	Rectangular, equal frequencies	Mean = Median
Skewed Left	Tail to the left	Mean < Median
Skewed Right	Tail to the right	Mean > Median