Describing, Exploring, and Comparing Data: Measures of Center

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 helps summarize and interpret 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:

$\bar{x} = \frac{\sum x}{n}$ (sample mean)

$\mu = \frac{\sum x}{N}$ (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: One 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 least to greatest.

• Calculation:

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

  • If the number of data values is 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 use every data value directly.

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 halfway between the maximum and minimum values in the data set.

• Formula:

$\text{Midrange} = \frac{\text{Maximum value} + \text{Minimum value}}{2}$

• 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

• Not all data sets are suitable for calculating measures of center. For example:

  • Zip codes, jersey numbers, or ranks are not quantitative measurements and do not have meaningful means or medians.

  • Top 5 or top 10 lists are not representative of the entire population.

  • Calculating the mean of means (e.g., state averages) without considering population sizes can be 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 frequency.

$\bar{x} = \frac{\sum (f \cdot x)}{\sum f}$

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

Weighted Mean

• When data values have different weights, the weighted mean is calculated as follows:

$\bar{x} = \frac{\sum (w \cdot x)}{\sum w}$

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

Examples

• Mean Example: For the wait times 50, 25, 75, 35, 50, 25, 30, 50, 45, 25, 20, the mean is 39.1 minutes.

• Median Example (Odd n): For the same wait times, sorted: 20, 25, 25, 25, 30, 35, 45, 50, 50, 50, 75. The median is 35.0 minutes.

• Median Example (Even n): Adding another value (50), sorted: 20, 25, 25, 25, 30, 35, 45, 50, 50, 50, 50, 75. The median is the mean of 35 and 45, which is 40.0 minutes.

• Mode Example: For the wait times 35, 35, 20, 50, 95, 75, 45, 50, 30, 35, 30, the mode is 35 minutes (occurs three times).

• Midrange Example: For the wait times 50, 25, 75, 35, 50, 25, 30, 50, 45, 25, 20, the midrange is (75 + 20) / 2 = 47.5 minutes.

• Weighted Mean Example (GPA): For grades A (3 credits), A (4 credits), B (3 credits), C (3 credits), F (1 credit), with quality points 4, 4, 3, 2, 0, the GPA is 3.07 (rounded to 3.1 or 3.07 as per convention).