Descriptive Statistics

Measures of Central Tendency

Measures of central tendency are values that represent a typical or central entry of a data set. They are essential for summarizing and understanding data by identifying the center point around which the data are distributed. The most common measures are the mean, median, and mode.

Mean

• Definition: The mean (or average) is the sum of all data entries divided by the number of entries.

• Population Mean Formula: where is the population mean, is the sum of all population data entries, and is the number of entries in the population.

• Sample Mean Formula: where is the sample mean, is the sum of all sample data entries, and is the number of entries in the sample.

• Example: For the sample weights 274, 235, 223, 268, 290, 285, 235, the mean is pounds.

Median

• Definition: The median is the value that lies in the middle of the data when the data set is ordered. It divides the data into two equal parts.

• Odd Number of Entries: The median is the middle data entry.

• Even Number of Entries: The median is the mean of the two middle data entries.

• Example (Odd): Ordered data: 223, 235, 235, 268, 274, 285, 290. Median is 268 (the fourth entry).

• Example (Even): Ordered data: 223, 235, 235, 268, 274, 290. Median is pounds.

Mode

• Definition: The mode is the data entry that occurs with the greatest frequency.

• No Mode: If no entry is repeated, the data set has no mode.

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

• Example (Numerical): In the data set 223, 235, 235, 268, 274, 285, 290, the mode is 235 (occurs twice).

• Example (Categorical): If the most frequent response in a survey is "Democrat," then the mode is "Democrat."

Comparing the Mean, Median, and Mode

• All three measures describe a typical entry of a data set.

• Mean: Takes every entry into account but is greatly affected by outliers (extreme values).

• Median: Not affected by outliers; often better represents the center when outliers are present.

• Mode: May not always represent a typical value, especially if the data set is uniform or multimodal.

• Example: In a class with ages mostly around 20 but one student aged 65, the mean is influenced by 65, but the median better represents the typical age.

Weighted Mean

• Definition: The mean of a data set whose entries have varying weights.

• Formula: where is the weight of each entry .

• Example: Calculating a grade point average (GPA) where each grade has a different credit hour weight.

Mean of Grouped Data

• Definition: The mean of a frequency distribution is estimated using class midpoints and frequencies.

• Formula: where is the class midpoint, is the frequency, and is the total number of data values.

• Example: If the mean screen time for 30 adults is estimated using grouped data, the result is an approximation based on midpoints.

The Shape of Distributions

The shape of a data distribution helps describe how data values are spread and where most values are concentrated.

• Symmetric Distribution: A vertical line can be drawn through the middle, creating mirror-image halves. The mean and median are approximately equal.

• Uniform Distribution (Rectangular): All entries or classes have equal or nearly equal frequencies. The distribution is symmetric.

• Skewed Left Distribution (Negatively Skewed): The "tail" of the graph elongates more to the left. The mean is to the left of the median.

• Skewed Right Distribution (Positively Skewed): The "tail" of the graph elongates more to the right. The mean is to the right of the median.

Additional info: The image included is the cover of the textbook "Elementary Statistics" by Ron Larson, which is directly relevant as it visually identifies the source of the material and reinforces the academic context of the notes.