Measures of Center in Descriptive Statistics

Describing, Exploring, and Comparing Data

Measures of Center

Measures of center are fundamental concepts in descriptive statistics, used to identify a central value that represents a data set. The most common measures include the mean, median, mode, and midrange. Understanding these measures helps in summarizing and interpreting data effectively.

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

Mean (Arithmetic Mean)

The mean, or arithmetic mean, is the sum of all data values divided by the number of values. It is widely used but sensitive to extreme values (outliers).

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

• Formula:

$x = \frac{\sum x}{n}$

• Sample Mean: Denoted by $\bar{x}$ (x-bar).

• Population Mean: Denoted by $\mu$ (mu).

• 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: For the data set 50, 25, 75, 35, 50, 25, 30, 50, 45, 25, 20, the mean is $\frac{430}{11} = 39.1$ 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.

• Definition: The median is the middle value in an ordered data set.

• Notation: Sometimes denoted by $\tilde{x}$ (x-tilde), M, or Med.

• 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 outliers.

  • Does not change much with extreme values.

Example (Odd): Sorted data: 20, 25, 25, 25, 30, 35, 45, 50, 50, 50, 75. Median is 35.0 minutes.

Example (Even): Sorted data: 20, 25, 25, 25, 30, 35, 45, 50, 50, 50, 75, 50. Median is $\frac{35+45}{2} = 40.0$ minutes.

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.

• Definition: The mode is the value(s) with the greatest frequency.

• Properties:

  • Can be found with qualitative data.

  • Data set can be unimodal, bimodal, multimodal, or have no mode.

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

Other Examples: Two modes: 30 and 50; No mode: 20, 30, 35, 50, 75 (no repeated values).

Midrange

The midrange is the value midway between the maximum and minimum values. It is sensitive to extremes and not resistant.

• Definition: Midrange is calculated as:

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

• Properties:

  • Easy to compute.

  • Reinforces that there are multiple ways to define center.

  • Rarely used in practice.

Example: For the data set 50, 25, 75, 35, 50, 25, 30, 50, 45, 25, 20, midrange is $\frac{75+20}{2} = 47.5$ minutes.

Round-Off Rules for Measures of Center

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

• For mode: leave the value as is, without rounding.

Critical Thinking in Measures of Center

It is important to consider whether calculating measures of center is meaningful for a given data set. The sampling method and the nature of the data should be evaluated before interpreting these statistics.

• Some data (e.g., zip codes, ranks, jersey numbers) are not suitable for calculating mean or median.

• Top lists (e.g., CEO compensation) may not represent the larger population.

• Weighted means may be necessary when population sizes differ.

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:

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

Example: For the frequency distribution:

Time (seconds)	Frequency (f)	Class Midpoint (x)	f × x
0 – 14	6	7.0	42.0
15 – 29	18	22.0	396.0
30 – 44	14	37.0	518.0
45 – 59	5	52.0	260.0
60 – 74	5	67.0	335.0
75 – 89	1	82.0	82.0
90 – 104	1	97.0	97.0
Totals	50		1730.0

Mean is $\frac{1730.0}{50} = 34.6$ minutes (approximate).

Weighted Mean

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:

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

Example: Calculating grade-point average:

• Grades: A (3 credits), A (4 credits), B (3 credits), C (3 credits), F (1 credit)

• Quality points: 4, 4, 3, 2, 0

• Weights: 3, 4, 3, 3, 1

• Calculation: $\frac{(3 \times 4) + (4 \times 4) + (3 \times 3) + (3 \times 2) + (1 \times 0)}{3+4+3+3+1} = \frac{43}{14} = 3.07$

Grade-point average is typically rounded to two decimal places (3.07 → 3.1).