Describing, Exploring, and Comparing Data

Measures of Center

Measures of center are statistical values that describe the central point or typical value of a data set. Understanding these measures is essential for summarizing and interpreting data effectively. The most common measures of center are the mean, median, mode, and midrange.

Mean (Arithmetic Mean)

• Definition: The mean is calculated by adding all data values and dividing by the number of values.

• Formula:

(sample mean)

(population mean)

• Properties:

  • Uses every data value in the calculation.

  • Sample means from the same population tend to vary less than other measures of center.

  • Not resistant: A single extreme value (outlier) can significantly affect the mean.

• Important Note: The term "average" is discouraged in statistics because it can refer to different measures of center.

Median

• Definition: The median is the middle value when the data set is ordered from smallest to largest.

• Calculation:

  • If the number of data values is odd, the median is the middle value.

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

• Properties:

  • Resistant to outliers: Extreme values do not significantly affect the median.

  • Does not directly use every data value in its calculation.

Mode

• Definition: The mode is the value(s) that occur(s) most frequently in a data set.

• Properties:

  • Can be used with qualitative (categorical) data.

  • A data set may have no mode, one mode (unimodal), two modes (bimodal), or more than two modes (multimodal).

Midrange

• Definition: The midrange is the value midway between the maximum and minimum values in the data set.

• Formula:

• Properties:

  • Very sensitive to extreme values (not resistant).

  • Rarely used in practice but easy to compute.

  • Helps illustrate that there are multiple ways to define the center of a data set.

Round-Off Rules for Measures of Center

• For mean, median, and midrange: Carry one more decimal place than the original data values.

• For mode: Do not round; report the value as it appears in the data set.

Critical Thinking: When Measures of Center Are Not Meaningful

• Measures of center are not always meaningful, especially for data that are labels, ranks, or not representative of the population.

• Examples:

  • Zip codes, jersey numbers, and ranks do not represent measurable quantities.

  • Top 5 CEO salaries are not representative of all CEO salaries.

  • Calculating the mean of means (e.g., state mean ages) without considering population sizes is misleading.

Calculating the Mean from a Frequency Distribution

When data are summarized in a frequency distribution, the mean can be approximated by multiplying each class midpoint by its frequency, summing these products, and dividing by the total number of data values.

• This method provides an approximation because it uses class midpoints instead of actual data values.

Weighted Mean

When data values have different levels of importance (weights), the weighted mean is used. Each value is multiplied by its weight, the products are summed, and the sum is divided by the total of the weights.

• Example: Calculating a grade-point average (GPA) where course grades are weighted by credit hours.