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Unit21.pdf
Unit 2: Basic Descriptive Statistics
Learning Objectives
Explain how descriptive statistics can make data understandable.
Construct and analyze frequency distributions for variables at each of the three levels of
measurement.
Compute and interpret percentages, proportions, ratios, rates, and percentage change.
Construct and analyze bar and pie charts, histograms, and line graphs.
Analyze bar and pie charts, histograms, and line graphs.
Use SPSS to generate frequency distributions and analyze the output.
Unit Outline
Using Statistics
Frequency Distributions for Variables at the Nominal Level
Enhancing Clarity: Percentages and Proportions
Frequency Distributions for Variables at the Ordinal Level
Frequency Distributions for Variables at the Interval-Ratio Level
Ratios, Rates, and Percentage Change
Using Graphs to Present Data
Using Statistics
Statistics can be used to summarize the scores on a single variable. Examples
include:
Expressing the percentage of teenagers who have been diagnosed with
Chlamydia.
Comparing feelings of ethnic and racial prejudice among college
students in the 1960’s to today.
Reporting changes in educational attainment over time.
Organizing information into easy-to-read tables, charts, and graphs.
Creating graphs to show changes in obesity rates over time.
Frequency Distributions
Introduction
Statistics can be used to summarize the scores on a single variable. Examples
include:
Expressing the percentage of teenagers who have been diagnosed with
Chlamydia.
Comparing feelings of ethnic and racial prejudice among college
students in the 1960’s to today.
Reporting changes in educational attainment over time.
Organizing information into easy-to-read tables, charts, and graphs.
Creating graphs to show changes in obesity rates over time.
Example of a table with nominal data
Gender distribution among volunteer workers at a blood donation center.
Value Frequency (f)
Male 16
Female 14
N=30
Percentages and Proportions
-Percentages standardize raw data to a base of 100.
Percentages range from 0 to 100.
Percentages can be used to compare data across samples of different
sizes.
Percentages are calculated by dividing a frequency (e.g. the number of
males) by the sample size (e.g. the total number of students) and then
multiplying by 100).
Proportions standardize raw data to a base of 1.00.
-Proportions standardize raw data to a base of 1.00.
Proportions range from 0 to 1.00.
Proportions can be used to compare data across samples of different
sizes.
Proportions are calculated by dividing a frequency (e.g. the number of
females) by the sample size (e.g. the total number of students).
Note: Don’t multiply by 100. That would give you a percentage.
100 N
f (%)Percentage
Example: Calculate the proportion of students in your statistics class
that have a part-time job.
-You will need to know the total number of students in the class (N) and the number of students
with part-time jobs (f).
Other points to keep in mind:
Percentages tend to be easier for most people to comprehend.
Proportions are more commonly used when working with probabilities
(see Chapter 5).
For smaller numbers of cases (less than 20), frequencies are more
commonly used than both percentages and proportions.
Always report the sample or population size along with percentages or
proportions to enable more meaningful interpretation.
Frequency Distribution for Variables Measured at the Ordinal Level
Example of a table with ordinal data
Worker satisfaction among volunteer workers at a blood donation center.
Value Frequency (f)
(1) Very Dissatisfied 10
(2) Dissatisfied 5
(3) Satisfied 8
N
f oportionPr
N
f oportionPr
(4) Very Satisfied 7
N=30
Example of a table with interval- Ratio data:
Age Distribution for volunteer workers at a blood donation clinic.
Value Frequency (f)
16 2
17 1
18 2
20 4
21 8
22 3
23 4
24 6
N=30
Constructing the Frequency Distribution
Class intervals are the categories listed in the table.
Stated limits are the listed categories, which sometimes appear to
have gaps between them.
-Midpoints are the points exactly halfway between the upper and lower limits of a class interval.
Cumulative frequency column lists the total number of cases falling
at a particular score or with a lower value in the distribution.
Cumulative percentage column lists the total percentage of cases
falling at a particular score or with a lower value in the distribution.
This frequency distribution contains unequal intervals. 18-19 contains
three years and 20-21 contains two years. It also contains an open-ended
interval (28 and older).
The presence of an unequal interval or an open-ended interval indicates that the variable is
measured at the ordinal level. The categories can be ranked from low to high, but we cannot
quantify the distance between the scores
As indicated above, this distribution is also measured at the ordinal level. It
contains both unequal intervals and an open-ended interval.
Which intervals are unequal? Which are open-ended?
Value Frequency (f)
Less than $10,000 8,329,488
$10,000 to $14,999 6,305,311
$15,000 to $24,999 12,172,059
$25,000 or more 858,041,717
N=884,848,575
Ratios, Rates and Percentage Changes
Ratios are useful for comparing the relative size of two different categories or
groups.
Ratios are calculated by dividing one comparison unit by another
comparison unit.
For clarity, the two comparison units (f 1
and f 2 ) are usually expressed.
For example, to express that there are 3 members of group 1 for every
1 member of group 2, you would write: 3:1.
Ratios Example: Calculate the ratio of juniors to seniors in your statistics
class.
You will need the number of juniors (f 1 ) and the number
of seniors (f 2 ) in your class.
2
1
f
f Ratio
2
1
f
f Ratio
Rates are yet another way of summarizing the distribution of a single variable.
Rates are computed by dividing the number of occurrences of
something (such as homicides) by the total number of possible
occurrences (such as the entire population of Los Angeles), and
multiplying by some power of 10.
Example: Calculate the homicide rate per 100,000.
You will need the number of homicides (for the
numerator), the total population (for the denominator),
and the base of 10 being used (100,000).
Percentage change can be used to measure the extent of an increase or a decrease
in a variable over time.
In the formula below, substitute scores (frequencies, rates, proportions of
percentages) with f 1 and f
2 where f
1 represents an earlier period in time and f
2 , a later
period of time.
Percentage change Example: Calculate the change in student enrollment at your
university from 1980 to 2010.
You will need the number of students in 1980 (f 1 ) and the
number of students in 2010 (f 2 ).
)10ofpower( soccurrencepossibleof#total
soccurrenceof# Rate
)000,100( populationtotal
icideshomof# RateHomicide
100 f
ff changePercent
1
12
Graphic Presentation of Data
Charts and graphs are used to present data in ways that are visually more
dramatic than frequency distributions.
Charts and graphs are particularly useful for conveying an impression of the overall
shape of a distribution or for highlighting any clustering of cases.
Pie charts are appropriate for variables at any level of measurement that have only a few
categories.
Pie charts display the percentages of each of a variable’s categories.
Bar charts are also appropriate for variables at any level of measurement that have
only a few categories.
Bar charts are particularly effective for displaying the relative frequencies for two
or more categories of a variable.
Histograms are used with interval-ratio level variables with many scores.
100 f
ff changePercent
1
12
Histograms are like bar charts, except that the sides of the “bars” touch to form a
continuous series.
Line charts (or frequency polygons) are also used with interval-ratio level
variables with many scores.
Line charts are like histograms, except that a dot is placed at the midpoint of a
category (instead of a bar), and those dots are connected by a line.
Summary
There are many statistics that can be used to summarize the distribution of a single
variable. In this chapter, we considered percentages, proportions, rates, ratios, and
percent changes.
Our emphasis was on the need to communicate our results clearly and concisely.
Frequency distributions summarize the entire distribution of a variable.
Columns for percentages, cumulative frequencies and/or cumulative percentages
can enhance the readability of frequency distributions.
Percentages and proportions report the relative occurrence of some category of a variable
compared with the distribution as a whole.
Ratios compare two categories with one another.
Rates report the actual occurrence of some phenomenon compared with the number of
possible occurrences per some unit of time.
Percentage change shows the relative increase or decrease in a variable over time.
Pie and bar charts, histograms, and line charts (or frequency polygons) are graphs used to
express the basic information contained in the frequency distribution in a compact and
visually dramatic way.
Basic Terms
Bar chart
Class intervals
Cumulative frequency
Cumulative percentage
Frequency distribution
Frequency polygon
Histogram
Line chart
Midpoint
Percentage
Percent change
Pie chart
Proportion
Rate
Ratio
Stated class limits
