Measures of Center: Describing, Exploring, and Comparing Data

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 in statistics.

• Key Concept: The main goal is to find and interpret values that represent the center of a data set, such as the mean and median.

Definition: Measure of Center

• A measure of center is a value at the center or middle of a data set.

Mean (Arithmetic Mean)

• The mean (or arithmetic mean) of a set of data is the measure of center found by adding all of the data values and dividing the total by the number of data values.

• Caution: The term average is often used for the mean, but it is not precise and is not used by statisticians or in professional journals.

Important Properties of the Mean

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

• The mean uses every data value in its calculation.

• The mean is not resistant: one extreme value (outlier) can substantially change its value.

Definition: Resistant Statistic

• A statistic is resistant if the presence of extreme values (outliers) does not cause it to change very much.

Formula for the Mean

• For a sample: $\bar{x} = \frac{\sum x}{n}$

• For a population: $\mu = \frac{\sum x}{N}$

• Where $\sum$ denotes the sum, $x$ represents individual data values, $n$ is the sample size, and $N$ is the population size.

Example: Calculating the Mean

• Given wait times (minutes): 50, 25, 75, 35, 50, 25, 30, 50, 45, 25, 20

• Calculation: $\bar{x} = \frac{50 + 25 + 75 + 35 + 50 + 25 + 30 + 50 + 45 + 25 + 20}{11} = \frac{430}{11} = 39.1$ minutes

Median

• The median is the middle value when the data are arranged in order of increasing (or decreasing) magnitude.

Important Properties of the Median

• The median is resistant to extreme values (outliers).

• The median does not directly use every data value; changing the largest value does not affect the median unless it changes the order of the middle value(s).

Calculation and Notation of the Median

• The median of a sample is sometimes denoted by $\tilde{x}$ ("x-tilde"), $M$, or Med.

• To find the median, first sort the values, then:

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

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

Example: Median with Odd Number of Data Values

• Data: 50, 25, 75, 35, 50, 25, 30, 50, 45, 25, 20

• Sorted: 20, 25, 25, 25, 30, 35, 45, 50, 50, 50, 75

• Median: 35.0 minutes (middle value of 11 data points)

Example: Median with Even Number of Data Values

• Data: 50, 25, 75, 35, 50, 25, 30, 50, 45, 25, 20, 50

• Sorted: 20, 25, 25, 25, 30, 35, 45, 50, 50, 50, 50, 75

• Median: $\frac{35 + 45}{2} = 40.0$ minutes (mean of the two middle values)

Mode

• The mode is the value(s) that occur(s) with the greatest frequency in a data set.

Important Properties of the Mode

• The mode can be found with qualitative (categorical) data.

• A data set can have no mode, one mode, or multiple modes.

Finding the Mode

• If two values occur with the same greatest frequency, the data set is bimodal.

• If more than two values occur with the same greatest frequency, the data set is multimodal.

• If no value is repeated, there is no mode.

Example: Mode

• Data: 35, 35, 20, 50, 95, 75, 45, 50, 30, 35, 30

• Sorted: 20, 30, 30, 35, 35, 35, 45, 50, 50, 75, 95

• Mode: 35 minutes (occurs three times)

Other Mode Examples

• Two modes: Data: 30, 30, 50, 50, 75 → Modes: 30 and 50

• No mode: Data: 20, 30, 35, 50, 75 → No value is repeated

Midrange

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

• Formula: $\text{Midrange} = \frac{\text{maximum data value} + \text{minimum data value}}{2}$

Important Properties of the Midrange

• The midrange is very sensitive to extreme values (not resistant).

• It is rarely used in practice, but it is easy to compute and helps illustrate different ways to define the center of a data set.

• Sometimes confused with the median, so clear definitions are important.

Example: Midrange

• Data: 50, 25, 75, 35, 50, 25, 30, 50, 45, 25, 20

• Calculation: $\text{Midrange} = \frac{75 + 20}{2} = 47.5$ minutes

Round-Off Rules for Measures of Center

• For the mean, median, and midrange, carry one more decimal place than is present in the original data values.

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

Critical Thinking: When Measures of Center Are Not Meaningful

• Always consider whether it makes sense to calculate a measure of center for a given data set.

• Consider the sampling method used to collect the data.

Examples Where Mean and Median Are Not Meaningful

• Zip codes: Numbers are just labels, not measurements.

• Ranks: Reflect order, but not measurable quantities.

• Jersey numbers: Arbitrary labels, not measurements.

• Top 5 CEO salaries: Not representative of the larger population.

• Mean of state means: Does not account for different population sizes; use a weighted mean instead.