Descriptive Statistics: Measures of Central Tendency

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

Definition and Importance

Measures of central tendency are statistical values that represent a typical or central entry of a data set. They are used to summarize and describe the center point of a distribution of data.

• Mean

• Median

• 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 of central tendency and is sensitive to 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 taken from the population.

Formulas:

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

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

Example: For the sample weights (in pounds): 274, 235, 223, 268, 290, 285, 235 Sum: $\sum x = 1810$ Sample size: $n = 7$ Mean: $\bar{x} = \frac{1810}{7} \approx 258.6$ pounds

Median

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

• If the data set has an odd number of entries, the median is the middle entry.

• If the data set has an even number of entries, the median is the mean of the two middle entries.

Example: For the ordered weights: 223, 235, 235, 268, 274, 285, 290 Median (odd number of entries): 268 pounds (the fourth entry)

Example (even number): If the data set is 235, 235, 268, 274, 290, 285 Median: Mean of 268 and 274 = $\frac{268 + 274}{2} = 271$ pounds Additional info: The actual example in the notes uses 268 and 274, but the sum and division may differ based on the actual data.

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, the data set has no mode.

• If two entries occur with the same greatest frequency, both are modes (bimodal).

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

Example (categorical data): In a survey of political party affiliation, if 'Democrat' is the most frequent response, then the mode is 'Democrat'.

Comparing Mean, Median, and Mode

Each measure describes a typical entry of a data set, but they have different properties and are affected differently by the distribution of data.

• Mean: Takes every entry into account; sensitive to outliers.

• Median: Less affected by outliers; represents the center of ordered data.

• Mode: May not always represent a typical value, especially in data sets with no repeated values or with multiple modes.

Example: In a class with ages: 20, 20, 20, 21, 21, 22, 22, 23, 65

• Mean: $\approx 23.8$ 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 value if the data is multimodal or if the most frequent value is not central.

Weighted Mean

The weighted mean is used when data entries have different weights or importance. It is commonly used in calculating grade point averages and other situations where values contribute unequally.

Formula:

• $\bar{x} = \frac{\sum xw}{\sum w}$, where $x$ is the value and $w$ is its weight.

Example: If grades and credit hours are given, multiply each grade by its credit hours, sum the products, and divide by the total credit hours.

Mean of Grouped Data

When data is presented in a frequency distribution, the mean can be approximated using class midpoints and frequencies.

• Find the midpoint ($x$) of each class: $x = \frac{\text{Lower limit} + \text{Upper limit}}{2}$

• Multiply each midpoint by its frequency ($f$), sum the products: $\sum xf$

• Sum the frequencies: $n = \sum f$

• Calculate the mean: $\bar{x} = \frac{\sum xf}{n}$

Example: If the total of $\sum xf = 8631$ and $n = 30$, then $\bar{x} = \frac{8631}{30} \approx 287.7$ minutes.

Additional info: This method provides an estimate because it uses class midpoints rather than actual data values.

Shape of Distributions

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

• Symmetric Distribution: The left and right halves of the graph are approximately mirror images. Mean, median, and mode are equal or nearly equal.

• Uniform Distribution: All classes have equal or nearly equal frequencies; the distribution is flat and 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.

Additional info: Graphical representations can help determine which measure of central tendency best describes the data set.