Describing, Exploring, and Comparing Data: Measures of Center

Describing, Exploring, and Comparing Data

Measures of Center

Measures of center are fundamental concepts in statistics, used to identify a central value within a data set. The most common measures include the mean, median, mode, and midrange. Understanding these measures allows statisticians to summarize and interpret data effectively.

• Measure of Center: A value representing the middle or center of a data set.

Mean (Arithmetic Mean)

The mean is calculated by adding all data values and dividing by the number of values. It is widely used but sensitive to extreme values (outliers).

• Formula: For a sample, the mean is denoted by $\bar{x}$; for a population, it is denoted by $\mu$.

• Equation: $\bar{x} = \frac{\sum x}{n}$ (sample mean), $\mu = \frac{\sum x}{N}$ (population mean)

• Properties:

  • Uses every data value.

  • Sample means vary less than other measures of center.

  • Not resistant to outliers.

• Caution: The term "average" is not used by statisticians for measures of center.

Example: The mean of wait times for “Space Mountain” is calculated as $\bar{x} = 39.1$ minutes.

Median

The median is the middle value when data are arranged in order. It is resistant to outliers and does not use every data value directly.

• Calculation:

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

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

• Properties: Resistant to extreme values.

Example: For eleven wait times, the median is 35.0 minutes. For twelve wait times, the median is the mean of the two middle values (35 and 45).

Mode

The mode is the value(s) that occur most frequently in a data set. It can be used with qualitative data and a data set may have no mode, one mode, or multiple modes.

• Types:

  • Bimodal: Two values occur with the same greatest frequency.

  • Multimodal: More than two values occur with the same greatest frequency.

  • No mode: No value is repeated.

Example: The mode of wait times for “Tower of Terror” is 35 minutes (occurs three times).

Midrange

The midrange is the value midway between the maximum and minimum values. It is calculated as:

• Formula: $\text{Midrange} = \frac{\text{Maximum} + \text{Minimum}}{2}$

• Properties: Not resistant to outliers; rarely used in practice but easy to compute.

Example: For eleven wait times, the midrange is 47.5 minutes.

Round-Off Rules for Measures of Center

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

• For mode: leave the value as is.

Critical Thinking: When Measures of Center Are Not Meaningful

It is important to consider whether calculating measures of center makes sense for a given data set. Some examples where mean and median are not meaningful include:

• Zip codes (just labels, not measurements)

• Ranks (reflect ordering, not measurement)

• Jersey numbers (labels, not measurements)

• Top 5 CEO compensation (not representative of the population)

• Mean ages of states (population sizes must be considered; use weighted mean)

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 the products, and dividing by the total frequency.

• Formula: $\bar{x} = \frac{\sum (f \cdot m)}{\sum f}$, where $f$ is frequency and $m$ is class midpoint.

• This method provides an approximation.

Calculating a Weighted Mean

A weighted mean is used when data values have different weights. It is calculated by multiplying each value by its weight, summing the products, and dividing by the sum of the weights.

• Formula: $\bar{x}_w = \frac{\sum (w \cdot x)}{\sum w}$

Example: Calculating grade-point average using course credits as weights and grade points as values. The result is a GPA of 3.07 (rounded to 3.1).