copyright © 2012 by nelson education limited.2-1 chapter 2 basic descriptive statistics:...

Copyright © 2012 by Nelson Education Limited. 2-1

Chapter 2Basic Descriptive Statistics: Percentages, Ratios

and Rates, Tables, Charts, and Graphs


In this presentation you will learn about:

• Percentages and Proportions

• Ratios and Rates

• Frequency Distributions for Variables Measured at the Nominal and Ordinal Levels

• Frequency Distributions for Variables Measured at the Interval-Ratio Level

• Charts and Graphs


Percentages and Proportions

n

f(p) Proportion

100=(%) Percentage

n

f

where f = frequency, or number of cases in any category (i.e., the part)n = number of cases in all categories (i.e., the whole)


• Question: What percentage of females are social science majors?

• Answer: the whole (n) is the number of females. n = 698

the part (f) is the number of social science majors within the female group. f = 246

therefore, (246/698)*100 = (.3524)*100 = 35.24%

Percentages and Proportions: An Example


Percentages and Proportions: An Example (continued)


• A Ratio compares the relative sizes of categories; that is: part to part.– Ratio = f1 / f2

• f1 - number of cases in first category

• f2 - number of cases in second

category

Ratios


• In a classroom of 42 students, there are 23 females and 19 males. So,

– The ratio of males to females is: 19/23 = 0.83

• For every female, there are 0.83 males.

Or stated differently,– The ratio of females to males is: 23/19 =

1.21• For every male, there are 1.21 females.

Ratios: Examples


• A Rate is the number of actual occurrences of an event divided by the number of possible occurrences, per some unit of time.

– Note, a rate is usually multiplied by some power of 10 (e.g., 1,000; 100,000) to eliminate decimal points.

Rates


• The Crude Death Rate (CDR) for a population is defined as the number of deaths in that population (actual occurrences) divided by the number of people in the population (possible occurrences), per year.– The CDR is then multiplied by 1000:

– As an example, in 2006, Ontario had:

1. 84,524 deaths AND a total population of 12,160,282

(note, this is the estimated population on July 1, 2006).

2. Therefore, there were 6.95 deaths per 1,000 population in 2006

[(84,524 / 12,160,282) x 1,000 = 6.95]

Rates: An Example


• Summarize distribution of a variable by reporting the number of times each score of a variable occurred.

• General Rule for categories of frequency distribution:– Exhaustive: there must be enough categories so that

all observations fall into some category. – Mutually exclusive: the categories must be distinct so

that an observation will fall into only one category.

Frequency Distributions


Frequency Distributions for Nominal/Ordinal Level Variables


• Basic Consideration:– Large number of scores

• Requires collapsing or grouping of categories• Decide how many categories and how wide

Frequency Distributions for Interval-Ratio Level Variables


Constructing Frequency

Distributions for Interval-Ratio

Level Variables


Cumulative Frequency and Percentage Distributions show how many cases fall below a given score or class interval in the distribution.

Cumulative Frequency Distribution

Cumulative Percentage Distribution


The purpose of Real Limits is eliminate the “gap” between Stated Limits. This is necessary in constructing some graphs, such as the histogram

State Limits and Real Limits


Graphs and charts present frequency distributions graphically.

Graphs and Charts


Pie Chart:Marital Status of

Respondents (n = 20)


Bar Chart:Marital Status of

Respondents (n = 20)


Histogram: Hours Studied for Final Exam

(n = 105)

0

3

6

9

12

15

18

3 6 9 12 15 18 21 24 27 30 33 36

Freq

uenc

y

Study Hours


• Array the real limits of the intervals or scores along the horizontal axis (abscissa).

• Array frequencies along the vertical axis (ordinal).

• For each category, construct a bar with height corresponding to number of cases and width corresponding to real limits of intervals.

Constructing a Histogram from a Frequency

Distribution


A Frequency Polygon uses a dot (as opposed to a bar used in the Histogram) to represent the frequency of each real or stated interval. A line then connects the dots.

Frequency Polygon Hours Studied for Final Exam

(n = 105)

0

3

6

9

12

15

18

3 6 9 12 15 18 21 24 27 30 33 36

Freq

uenc

y

Study Hours

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