Ch. 3- Measures of Central Tendency: Mean, Median, and Mode

Sec 3.1 Measures of Central Tendency

Introduction to Measures of Central Tendency

Measures of central tendency numerically describe the average or typical value in a data set. The three most widely-used measures are the mean, median, and mode. These measures can yield different results depending on the data's distribution and characteristics.

• Mean: Commonly referred to as the "average" in media and everyday language, but sometimes the term "average" is used for median or mode.

• Median: The middle value when data are arranged in order.

• Mode: The most frequently occurring value in the data set.

Additional info: These measures help summarize large data sets and are foundational in statistical analysis.

Types of Measures for Central Tendency

• Population Mean ($\mu$): The mean of all values in a population. It is a parameter (a fixed value describing the population).

• Sample Mean ($\bar{x}$): The mean of values in a sample. It is a statistic (an estimate based on a sample).

• Mode: The value that appears most frequently in the data set.

• Median: The value that lies in the middle of the data when arranged in ascending order.

Calculating the Mean

The mean is calculated by adding all the values and dividing by the number of observations.

• Population Mean Formula:

$Mean = \frac{x_1 + x_2 + \cdots + x_N}{N} = \frac{\sum x_i}{N}$

• Example: For a sample of 50 households, the mean number of cars is calculated by summing all car counts and dividing by 50.

Computing a Population Mean and a Sample Mean

Given travel times (in minutes) for 7 employees: 23, 36, 23, 18, 5, 26, 43.

• Population Mean Calculation:

$Mean = \frac{23 + 36 + 23 + 18 + 5 + 26 + 43}{7} = \frac{174}{7} = 24.9$

• Sample Mean: Select a random sample of 3 employees, e.g., 5, 26, 36.

$\bar{x} = \frac{5 + 26 + 36}{3} = \frac{67}{3} = 22.3$

• Another sample: 36, 23, 26.

$\bar{x} = \frac{36 + 23 + 26}{3} = \frac{85}{3} = 28.3$

• Key Point: Sample means can underestimate or overestimate the population mean depending on the sample chosen.

Finding the Median

The median is the middle value in an ordered data set.

• If the number of data values ($n$) is odd, the median is the value at position $(n+1)/2$.

• If $n$ is even, the median is the mean of the two middle values.

Example: For the data 5, 18, 23, 23, 26, 36, 43 ($n=7$):

$Median =$ value at position $\frac{7+1}{2} = 4$ (the 4th value, which is 23).

If a new value (70) is added ($n=8$):

$Median =$ mean of values at positions 4 and 5: $\frac{23 + 26}{2} = 24.5$

• Interpretation: The median divides the data into two halves, with at most half the values below and half above.

Finding the Mode

The mode is the value that occurs most frequently in a data set.

• A data set can have one mode (unimodal), more than one mode (bimodal, multimodal), or no mode (if all values are unique).

• Example: Data: 1.10, 0.42, 0.73, 0.48, 1.10. Mode is 1.10.

• Example: Data: 27, 55, 55, 55, 88, 80, 99. Bimodal: 27 & 55.

• Example: Data: 6, 7, 8, 9, 10. No mode.

Comparing Mean and Median

Mean and median can give different results, especially when data are skewed or contain outliers.

• Mean: Sensitive to extreme values (not resistant).

• Median: Resistant to outliers and skewed data.

• Example: Adding a new employee with a 130-minute commute increases the mean significantly, but the median changes less.

Additional info: Use the median for skewed distributions or when outliers are present; use the mean for symmetric distributions.

Summary Table: Measures of Central Tendency

Measure	Definition	Best Used When
Mean	Sum of all values divided by number of values	Data are quantitative and distribution is roughly symmetric
Median	Middle value in ordered data	Data are quantitative and distribution is skewed or has outliers
Mode	Most frequent value	Data are qualitative or when the most frequent value is desired