1. 1Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Chapter 22
Using Statistics To Describe
Variables

2. 2Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Using Statistics to Describe
Variables
 Two major classes of statistics
 Descriptive statistics
• To reveal characteristics of the sample dataset
 Inferential statistics
• To gain information about effects in the population being
studied

3. 3Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Using Statistics to Describe
 All quantitative research uses descriptive
statistics
 For description of the sample
 For initial description of variables
 For analysis of the primary research problem
 Descriptive statistics for descriptive research
 Inferential statistics for interventional and
correlational research

4. 4Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Using Statistics to Summarize
Data
 Terms: the number of elements in a sample is
the “n” of the sample
 Data set: 45, 26, 59, 51, 42, 28, 26, 32, 31, 55, 43,
47, 67, 39, 52, 48, 36, 42, 61, 57
 n = 20
 Descriptive statistics
 Frequency distributions
 Measures of central tendency
 Measures of dispersion

5. 5Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Frequency Distributions
 Table or figure (line graph, pie chart, etc.)
 Continuous variable: the higher numbers
represent more of that variable, and the lower
numbers represent less of that variable

6. 6Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Frequency Table
 Listing every possible value in the first
column of numbers, and the frequency (tally)
of each value as the second column of
numbers
 Data set: 45, 26, 59, 51, 42, 28, 26, 32, 31,
55, 43, 47, 67, 39, 52, 48, 36, 42, 61, 57
(ages)
 Sort from lowest to highest values
 Tally each value

7. 7Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Ungrouped Frequency
Distribution
 List all categories of the variable
on which they have data, and tally
each datum on the listing
Age Frequency
26 2
28 1
31 1
32 1
36 1
39 1
42 2
43 1
45 1
47 1
48 1
51 1
52 1
55 1
57 1
59 1
61 1
67 1

8. 8Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Grouped Frequency Distribution
 Categories are grouped into ranges
 Ranges must be mutually exhaustive and
mutually exclusive
Age Frequency
20 - 29 3
30 - 39 4
40 - 49 6
50 - 59 5
60 - 69 2

9. 9Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Grouped Frequency Distribution
with Percentages
Adult Age
Range
Frequency
(f)
Percentage (%)
Cumulative
Percentage
20 – 29 3 15 15
30 – 39 4 20 35
40 – 49 6 30 65
50 – 59 5 25 90
60 – 69 2 10 100
Total 20 100

10. 10Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Frequency Distributions
Presented in Figures
 Graphs
 Charts
 Histograms
 Frequency polygons

11. 11Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Line Graph

12. 12Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Frequency Table of Smoking
Status
Smoking Status Frequency Percent
Current smoker 1 10
Former Smoker 6 60
Never Smoked 3 30
Total 10 100

13. 13Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Histogram of Smoking Status

14. 14Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Measures of Central Tendency
 Statistics that provides the center or hallmark
value of a data set
 Mode
 Median (MD)
 Mean

15. 15Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Mode
 The most common value in a data set
 Bimodal: two modes exist
 Multimodal: more than two modes

16. 16Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Median (MD)
 The middle value in the data set (after sorting
values from lowest to highest)
 If the “n” is even, the two values in the middle
are averaged
 The 50th percentile

17. 17Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Mean
 Arithmetic average of all a variable's values
 Most commonly reported measure of central
tendency
 Sum of the scores divided by the number of
values in the data set
 Formula:

18. 18Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
When to Use Mean
 Mean: normally distributed values measured
at the interval or ratio level
 Ordinal level data from a rating scale If
 The n is large
 The data are normally distributed
 Small values denote very little of the measured
quantity; large ones denote a lot
 Mean is sensitive to extreme scores such as
outliers

19. 19Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
When to Use Median and Mode
 Median: used for non-normal distributions
with small n
 Mode: used for nominal values

20. 20Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Using Statistics to Explore
Deviations in the Data
 Using measures of central tendency to
describe the nature of a data set obscures
the impact of extreme values or deviations in
the data
 Measures of dispersion, provide important
insight into nature of the data

21. 21Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Measures of Dispersion
 Quantifications of how tightly clustered
around the mean the sample is:
 Tightly clustered = fairly homogeneous
 Widely dispersed = heterogeneous
 Range
 Difference score
 Variance
 Standard deviation

22. 22Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Range
 Presented in two ways:
 The lowest score and the highest score (2 through 17)
 The difference between the highest and the lowest
score (range of 15)

23. 23Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Difference Score
 Subtract the mean from each score
 Sometimes referred to as a deviation score
 The difference score is positive when score is
above the mean, and negative when score is
below the mean
 The total of all the difference scores is zero
 Formula:

24. 24Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Mean Deviation
 Average difference score, using the absolute
values
 Example:

25. 25Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Variance (s²)
 Variance commonly used
 “s2”
is used to represent a sample variance
 “σ2”
is used to represent population variance
 Always a positive value, has no upper limit
 Bigger variances = more spread
 Formula:

26. 26Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Standard Deviation (s)
 Square root of the variance
 Sometimes reported as SD
 Most commonly reported measure of
dispersion
 Formula:

27. 27Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
The Normal Curve

28. 28Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
The Normal Curve (Cont’d)
 Represents the frequency distribution of a variable
that is perfectly normally distributed
 Signifies:
 The mean is the most commonly occurring value
 There are just as many values above the mean as there are
below the mean
 When frequency table is constructed, values are perfectly
symmetric
 68% of values are –1 to +1 standard deviations from mean
 95% of values are –2 to +2 standard deviations from mean

29. 29Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
z-Score

30. 30Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
z-Score (Cont’d)
 Synonymous with a standard deviation unit
 A z value of 1.0 represents 1 standard deviation
unit above the mean
 A z value of –1.0 represents 1 standard deviation
unit below the mean
 Formula:

31. 31Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Sampling Error
 Described by the statistic “standard error”
 Standard error of the mean is calculated to determine
the magnitude of the variability associated with the
mean
 Formula:
where
 = standard error of the mean
 s = standard deviation
 n = sample size

32. 32Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Confidence Interval
 Determines how closely a sample value
approximates a population value
 Can be created for many statistics, such as a
mean, proportion, and odds ratio
 Using a table of statistical values, the t-value
is accessed, for the desired interval, usually
95%

33. 33Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Confidence Interval (Cont’d)
 To calculate a 95% confidence interval
around a mean, for example:
 Calculate the mean
 Calculate the standard error of the mean
 Calculate the degrees of freedom (df) [df = n – 1]
 Look up the two-tailed t-value for p < 0.05

34. 34Copyright © 2013, 2009, 2005, 2001, 1997 by Saunders, an imprint of Elsevier Inc.
Degrees of Freedom
 The number of independent pieces of
information that are free to vary
 For confidence interval, the degrees of
freedom (df) are n – 1
 This means that there are n – 1 independent
observations in the sample that are free to vary (to
be any value) to estimate the lower and upper
limits of the confidence interval