Descriptive Statistics: Measures of Center and Spread (Chapter 3 Study Notes)

Descriptive Statistics

Overview

Descriptive statistics are numerical methods used to summarize and describe the main features of a data set. These statistics help characterize the data's center and spread, providing insight into its distribution and variability.

• Numbers that describe data

• Characterize the data’s center and spread

Notation

Statistical Symbols

Statistical notation is essential for understanding formulas and calculations in statistics. Common symbols include:

• Σ - denotes the sum of a set of values.

• x - represents individual values.

• n - number of values in a sample (sample size).

• N - number of values in a population (population size).

Population and Sample Size

Identifying Population and Sample

In statistics, it is important to distinguish between the population (entire group) and the sample (subset of the population). For example:

• Population size: All 50 states in a study.

• Sample size: 10 states selected from the population.

Measures of Center

Definition and Types

Measures of center describe the middle or central value of a data set. Common measures include the mean, median, mode, and midrange.

• Mean - arithmetic average

• Median - middle value

• Mode - most frequent value

• Midrange - average of maximum and minimum values

Mean

The mean is calculated by adding all values and dividing by the number of values. It is the most common measure of center.

• Sample mean: $\bar{x} = \frac{\Sigma x}{n}$

• Population mean: $\mu = \frac{\Sigma x}{N}$

• Note: Calculators typically use $\bar{x}$ for both sample and population mean.

Example: Calculating Mean

• Sample: 12, 25, 38, 5, 20

• Population: 60, 73, 77, 51, 68, 75

Calculator Notation

Statistical calculators use specific symbols for common statistics:

• $\bar{x}$: mean (average)

• $\Sigma x$: sum of elements

• $\Sigma x^2$: sum of squares

• $Sx$: sample standard deviation

• $\sigma x$: population standard deviation

• minX: minimum value

• Q1: first quartile

• Med: median

• Q3: third quartile

• maxX: maximum value

Mean of Grouped Data

For grouped data, the mean is calculated using class midpoints and frequencies:

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

• x: midpoint of each class

• f: frequency of each class

Example: Frequency Distribution

Find the mean of the frequency distribution and histogram:

Weight (lbs)	Frequency
26-95	5
96-165	6
166-235	7
236-305	5
306-375	4
376-445	2

Weighted Mean

The weighted mean accounts for varying importance (weights) of each entry:

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

• x: each entry

• w: weight of each entry

Example: Weighted Mean Calculation

Assignment and Grade	Weight
Test 1: 92	15%
Test 2: 94	15%
Test 3: 98	15%
Final Exam: 87	25%
Homework Average: 98	10%
Quiz Average: 84	10%
Project: 84	5%
Attendance: 89	5%

Resistance

The mean is sensitive to extreme values (outliers) and is not a resistant measure of center. This means that a single unusually high or low value can significantly affect the mean.

Median

The median is the middle value when data are arranged in order. It is denoted by Med and is a more resistant measure of center, as it is less affected by extreme values.

Example: Calculating Median

• 6.) 5.40, 1.10, 0.42, 0.66, 0.48, 1.10, 0.73

• 7.) 12, 22, 49, 80, 98, 141, 215, 239

Shape of the Distribution

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

• If perfectly symmetric, mean and median are equal.

• If skewed, mean is closer to the tail; median remains closer to the center.

• Median is preferred for skewed data.

Example: Mean vs. Median Resistance

Find the mean and median for the data set: 101, 106, 109, 116, 118, 135, 568. Contrast their resistance to outliers.

Mode and Modal Class

The mode is the value(s) that occur with the greatest frequency. Modal class refers to the class(es) with the highest frequency in grouped data. Mode is the only measure of center usable with categorical (qualitative) data.

Example: Finding Mode

• 9.) red, blue, purple, yellow, blue, green

• 10.) 17, 12, 18, 17, 21, 17, 19, 17

• 11.) Use the graph below (age of first visit to the coast)

Numbers of Modes

A data set can have one, more than one, or no mode:

• Unimodal: one mode

• Bimodal: two modes

• Multimodal: more than two modes

• No Mode: all values occur with equal frequency

Midrange

The midrange is the value midway between the maximum and minimum data values:

• $\text{midrange} = \frac{\text{max} + \text{min}}{2}$

• Since it uses extreme values, it is not a resistant measure.

Example: Calculating Midrange

• 12.) 31, 19, 78, 65, 95

• 13.) 932, 906, 117, 664, 144, 906, 626, 474

Measures of Spread

Definition and Types

Measures of spread describe the variation or dispersion of a data set. Common measures include range, standard deviation, and variance.

• Range - difference between maximum and minimum values

• Standard Deviation - average distance from the mean

• Variance - average squared distance from the mean