Measures of Center in Descriptive Statistics

Descriptive Statistics

Overview of Descriptive and Inferential Statistics

Descriptive statistics involve summarizing or describing the important characteristics of a data set, such as measures of center and variation. Inferential statistics use sample data to make inferences or generalizations about a population, including hypothesis testing, estimation, and prediction.

Populations, Samples, Parameters, and Statistics

Key Definitions

• Population: The complete collection of all elements to be studied.

• Sample: A subset of the population, selected for analysis.

• Parameter: A numerical measurement describing a characteristic of a population.

• Statistic: A numerical measurement describing a characteristic of a sample.

Good sampling methods include simple random samples and stratified samples, which help ensure that the sample represents the population accurately.

Measures of Center

Introduction to Measures of Center

Measures of center are values that describe the central or typical value in a data set. The most common measures are the mean, median, mode, and midrange.

Mean (Arithmetic Mean)

The mean is calculated by adding all data values and dividing by the number of values. It is commonly referred to as the "average." The mean is sensitive to extreme values (outliers).

• Sample mean: Denoted as $\bar{x}$ ("x-bar").

• Population mean: Denoted as $\mu$ ("mu").

Formula for the mean:

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

where $\sum x$ is the sum of all data values and $n$ is the number of values.

Example: Find the mean of the data set: 22, 22, 26, 24, 23.

Solution: $\bar{x} = \frac{22 + 22 + 26 + 24 + 23}{5} = \frac{117}{5} = 23.4$

• Advantages: Uses all data values; sample means vary less than other measures.

• Disadvantages: Not resistant to outliers; a single extreme value can affect it dramatically.

Median

The median is the middle value when the data are arranged in order. It is a resistant measure of center, meaning it is not affected much by extreme values.

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

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

Example: Data: 0.42, 0.48, 0.66, 0.73, 1.10, 1.10, 5.40 (odd number of values). Median is 0.73.

Data: 0.42, 0.48, 0.73, 1.10, 1.10, 5.40 (even number of values). Median is $(0.73 + 1.10)/2 = 0.915$.

Mode

The mode is the value that occurs with the greatest frequency in a data set. A data set may have one mode (unimodal), more than one mode (bimodal or multimodal), or no mode if all values are unique. The mode is the only measure of central tendency that can be used with nominal data.

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

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

• No mode: No value is repeated.

Example:

• Data: 5.40, 1.10, 0.42, 0.73, 0.48, 1.10 → Mode is 1.10

• Data: 27, 27, 27, 55, 55, 55, 88, 88, 99 → Bimodal (27 & 55)

• Data: 1, 2, 3, 6, 7, 8, 9, 10 → No mode

Midrange

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

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

The midrange is sensitive to extreme values and is rarely used as the sole measure of center.

Estimating the Mean from a Frequency Distribution

Frequency Distribution Mean

When data are grouped into classes, the mean can be estimated by assuming all values in a class are equal to the class midpoint. The formula is:

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

where $f$ is the frequency of each class and $x$ is the class midpoint.

Example: Estimating the mean from IQ scores:

IQ Score	Frequency f	Class Midpoint x	f · x
50-69	2	59.5	119.0
70-89	33	79.5	2623.5
90-109	35	99.5	3482.5
110-129	7	119.5	836.5
130-149	1	139.5	139.5
Totals	78		7201.0

Estimated mean: $\bar{x} = \frac{7201.0}{78} \approx 92.32$

Weighted Mean

Definition and Calculation

A weighted mean is used when data values are assigned different weights. The formula is:

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

where $w$ is the weight assigned to each value $x$.

Example: Calculating a student's GPA:

• Swahili: A (3 credits), Intro Stats: A (4 credits), Chemistry: B (3 credits), History: C (3 credits), Biology: F (1 credit)

• Quality points: A=4, B=3, C=2, F=0

• Weighted mean: $\bar{x} = \frac{(3 \times 4) + (4 \times 4) + (3 \times 3) + (3 \times 2) + (1 \times 0)}{3+4+3+3+1} = \frac{37}{14} \approx 3.07$

Comparing Measures of Center in Different Distributions

Skewness and Measures of Center

The relationship between the mean, median, and mode depends on the shape of the data distribution:

• Positively skewed (right-skewed): Mean > Median > Mode

• Symmetric: Mean = Median = Mode

• Negatively skewed (left-skewed): Mean < Median < Mode

Choosing the Best Measure of Center

Guidelines

• Use the mean for symmetric distributions without outliers.

• Use the median for skewed distributions or when outliers are present.

• Use the mode for categorical (nominal) data.

• The midrange is rarely used due to its sensitivity to extremes.

Example: For the monthly number of surveys: 1, 8, 10, 15, 16, 22, 131, 20, 17, 15, 15, 5, the mean is affected by the outlier (131), so the median may be a better measure of center.