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The Information School of the University of Washington
LIS 570
Session 6.1
Univariate Data Analysis
The Information School of the University of Washington
Objectives: Have answers to
the following questions
• Why is the normal distribution important for
statistical analysis (the ones presented) to make
sense?
• What is the logic behind inferential statistics?
(On what theories is it based?)
• What is a Confidence Interval?
• In what ways can we summarize quantitative data?
• What are some visualization techniques to help us
summarize and make sense of data?
LIS 570
Univariate Analysis
Mason; p. 2
The Information School of the University of Washington
Agenda
•
•
•
•
•
•
Exercise: understand “the problem”
Vocabulary
Functions of statistics
When to use what type
Descriptive statistics
Inferential statistics
LIS 570
Univariate Analysis
Mason; p. 3
The Information School of the University of Washington
Why and What
• Why know statistics?
–
–
–
–
Informed consumer…
Informed user…
Informed professional…
…
• What is a statistic?
a descriptive summary (index) of a sample
LIS 570
Univariate Analysis
Mason; p. 4
The Information School of the University of Washington
Sample and Population
Sample
Population (Universe)
A set of observations, instances,
individuals drawn from a
population, usually intended
to represent the population in
a study
The totality of things we are
interested in (e.g., the population
of all students at the UW)
Population
Sample
Average = 4.5
Average = 4.55
statistic
parameter
A statistic is a characteristic of a sample, while the same characteristic, if
descriptive of a population, is called a population parameter.
LIS 570
Univariate Analysis
Mason; p. 5
The Information School of the University of Washington
2 major functions of statistics
• Help us describe characteristics of sample
– Descriptive statistics
– Procedures to summarize, organize, and
simplify data
• Help us describe characteristics of population
– Inferential statistics
– Techniques for studying samples, and then
make generalizations about the population
from which the samples were selected.*
* Source: Gravetter, F. J. and Wallnau, L. B. (2002). Essentials of Statistics for the Behavioral
Sciences. 4th edition. Pacific Grove, CA: Wadsworth, p. 5
LIS 570
Univariate Analysis
Mason; p. 6
The Information School of the University of Washington
Vocabulary
• Variable—characteristic which has more than one
value
– e.g., Sex—male, female; hours of work/week—
anything from 0 – 168
– Independent variable (X)—manipulated by the
researcher or believed to be the cause of…
– Dependent variable (Y)—variable observed to assess
the effect of the manipulation, or changes depending on
the independent variable
• Data—observations (measurements) taken on the
units of analysis
LIS 570
Univariate Analysis
Mason; p. 7
The Information School of the University of Washington
Choosing the Statistical Technique*
Specific research question or hypothesis
Determine # of variables in question
Univariate analysis
Bivariate analysis
Multivariate analysis
Determine level of
measurement of variables
Choose univariate
method of analysis
Choose relevant
descriptive statistics
Choose relevant
inferential statistics
LIS 570
Univariate Analysis
* Source: De Vaus, D.A. (1991) Surveys in Social Research.
Third edition. North Sydney, Australia: Allen & Unwin Pty
Ltd., p133
Mason; p. 8
The Information School of the University of Washington
What To Do with a Bunch of Numbers
• Organize the observations
• Interested primarily in normality and deviations from normality
• Examine
– Central tendency
– Dispersion
– Shape of distribution
• Visualization aids
–
–
–
–
–
–
–
Frequency distribution (percentile) tables and charts
Histograms
Bar & pie charts (nominal data)
Frequency polygon
Cumulative percentage curve
Stem and leaf diagrams
Box plots
LIS 570
Univariate Analysis
Mason; p. 9
The Information School of the University of Washington
Frequency Distributions
• Ungrouped frequency distribution
• A list of each of the values of the variable
• The number of times and/or the percent of times
each value occurs
• Grouped frequency distribution
• A table or graph
• Shows frequencies or percent for ranges of values
LIS 570
Univariate Analysis
Mason; p. 10
The Information School of the University of Washington
Frequency distributions
Include in frequency distribution tables:
–
–
–
–
–
–
–
Table number and title
Labels for the categories of the variables
Column headings
Total number of cases (N)
The number of missing cases
Source of the data
Footnotes to explain anomalies and notes
* Source: De Vaus, D.A. (1991) Surveys in Social Research. Third edition. North Sydney, Australia: Allen & Unwin
Pty Ltd., p133
LIS 570
Univariate Analysis
Mason; p. 11
The Information School of the University of Washington
Grouped frequency distribution
Table 1—Example of grouped frequency distribution
Real Limits*
Frequencies
(ƒ)
Cumulative
frequencies
(Cf)
Percent
(%)
Cumulative
Percent
9-10
8.5 - 10.4999
3
20
15
100
7-8
6.5 - 8.4999
4
17
20
85
5-6
4.5 - 6.4999
7
13
35
65
3-4
2.5 - 4.4999
4
6
20
30
1-2
0.5 - 2.4999
2
2
10
10
Total (N)
20
Score Range
(Your value
label)
Valid cases: 20
100
Missing cases: 0
Note 1: “Real limits” of a score extend from one-half of the smallest unit of measurement
below the value of the score to one half unit above.
Note 2: Percent (%) = (ƒ /N) * 100, Cumulative % = (Cf/N) * 100
LIS 570
Univariate Analysis
Mason; p. 12
20
15
Frequency
The Information School of the University of Washington
Histogram
10
5
0
47
52 57 62 67 72 77 82 87 92 97
Statistics exam scores
-The height of the bar corresponds to the frequency (ƒ)
-The width of the bar extends to the real limits of the score
-Used only on interval and ratio scales
-No space between bars (that’s a bar chart)
LIS 570
Univariate Analysis
score
intervals
ƒ
45-49
1
50-54
2
55-59
4
60-64
4
65-69
7
70-74
9
75-79
16
80-84
10
85-89
7
90-94
6
95-99
2
Mason; p. 13
The Information School of the University of Washington
What do graphs (histograms)
show?
• Normality (normal distributions) [Why are
normal distributions important?]
• Deviations from normality
–
–
–
–
Positive skewness
Negative skewness
Bimodality
And more…
LIS 570
Univariate Analysis
Mason; p. 14
symmetrical
Normal distribution:
symmetrical Bell-shaped
curve
asymmetrical
The Information School of the University of Washington
Shapes of distribution
Positively skewed:
tail on the right, cluster towards low
end of the variable
Bimodality: A double peak
Negatively skewed:
tail on the left, cluster towards highend of the variable
LIS 570
Univariate Analysis
Mason; p. 15
The Information School of the University of Washington
Central Tendency
• Central tendency is a single summary figure that
ideally, is the most representative value of all
values in the distribution.
• Used to describe “typical” or representative value
Mean (arithmetic mean), m
– Sum all the observations; divide by N: use for interval
variables when appropriate
– Median: Value that divides the distribution so that an
equal number of values are above the median and an
equal number below
– Mode: Value with the greatest frequency (uni-modal,
bi-modal, etc.)
LIS 570
Univariate Analysis
Mason; p. 16
The Information School of the University of Washington
Variability, dispersion, spread
• Why do we care about anything
besides central tendency?
• Variability refers to spread or
dispersion
• The extent to which a set of scores
scatter about or cluster together
• Measures of variability
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–
–
–
–
–
Range
Interquartile range
Sum-of-squares
Variance
Standard deviation
Kurtosis
Equal means, unequal variability
LIS 570
Univariate Analysis
Mason; p. 17
The Information School of the University of Washington
Kurtosis
Two distributions: the same mean & variance
Karl Pearson suggested names
• Longer tailed: leptokurtic
• Shorter tailed: platykurtic
http://members.aol.com/jeff570/k.html
LIS 570
Univariate Analysis
Mason; p. 18
The Information School of the University of Washington
Mode (Mo): most common value
• Best for nominal level data
• Cautions:
–
–
–
–
most common may not measure typicality
not sensitive to outliers (good and bad)
may be more than one mode
unstable from sample to sample
• Dispersion
– variation ratio (v)
• % of people not in the modal category
LIS 570
Univariate Analysis
Mason; p. 19
The Information School of the University of Washington
Median (Mdn): Even split of sample
• For interval or ratio data, good for skewed
distributions (mean would not be a good measure
of central tendency)
• Minimal calculation (need to know frequencies)
• Reasonably insensitive to outliers (as long as there
are only a few)
• Reasonably stable from sample to sample
• Example of ordinal variables
– people are ranked from low to high (e.g., height)
– median is the middle case
– the median category is the one to which the middle person belongs
LIS 570
Univariate Analysis
Mason; p. 20
The Information School of the University of Washington
Median– simple examples
–1234567
• Mdn = 4
– 1 2 3 5 6 7 9 13
• Mdn = 5.5
by interpolation between 5 & 6 (5+6)/2 = 11/2 = 5.5
LIS 570
Univariate Analysis
Mason; p. 21
The Information School of the University of Washington
Dispersion
• The nth percentile of a set of numbers is a
value such that n percent of the numbers fall
below it and the rest fall above.
– The median is the 50th percentile
– The lower quartile is the 25th percentile
– The upper quartile is the 75th percentile
• Summary of sample using 5 numbers:
median, mean, variance, and extremes
LIS 570
Univariate Analysis
Mason; p. 22
The Information School of the University of Washington
Dispersion
Bottom 25%
Lower
quartile
Interquartile
range
Median
Top 25%
Upper
quartile
LIS 570
Univariate Analysis
Mason; p. 23
The Information School of the University of Washington
Boxplot
Interquartile range (IQR)
Variable 1
Variable 2
Variable 3
4
6
8
10
LIS 570
Univariate Analysis
12
14
16
Mason; p. 24
The Information School of the University of Washington
Mean
• Uses the actual numerical values of the
observations
• Most stable from sample to sample
• Most common measure of center
• Makes sense only for interval or ratio data
• Frequently computed for ordinal variables as well
• Not a good representation of central tendency for
skewed samples
LIS 570
Univariate Analysis
Mason; p. 25
The Information School of the University of Washington
Mean--Dispersion
• The standard deviation and variance measure spread
about the mean as centre.
• Deviation: distance and direction from the mean
– Doesn’t work as a measure of variability because adds up to zero
(see next slide).
• Variance
– mean of the squared deviation scores (of the deviations of
observations from the mean).
• Standard deviation
– Conceptually: the typical distance of scores from the mean
– Technically: the square root of the variance
LIS 570
Univariate Analysis
Mason; p. 26
The Information School of the University of Washington
Example Data (6,7,5,3,4)
x=
6+7+5+3+4 =
5
– Variance (S2)
•
•
•
•
•
25
5
= 5
Calculate the mean for the variable
Take each observation and subtract the mean from it
Square the result from the above
Add (sum) all the individual results
Divide by n
LIS 570
Univariate Analysis
Mason; p. 27
The Information School of the University of Washington
Variance (s2)
Observation
x
6
7
5
3
4
Deviation Sq. deviation
x-x
(x - x)2
6-5 = 1
1
7-5 = 2
4
5-5 = 0
0
3-5 = -2
4
4-5 = -1
1
Sum = 10
Variance = sum of the sq deviations = 10 = 2
number of observations
5
LIS 570
Univariate Analysis
Mason; p. 28
The Information School of the University of Washington
Standard deviation (s)
• Square root of the variance 2 = 1.4
• An average deviation of the observations
from their mean
• Influenced by outliers
• Best used with symmetrical distributions
LIS 570
Univariate Analysis
Mason; p. 29
The Information School of the University of Washington
Summary
• Descriptive statistics – univariate analysis
(central tendency, frequency distribution, dispersion)
• Determine if variable is nominal, ordinal or
interval
• Nominal: frequency tables, mode
• Ordinal
–
–
–
–
Frequency tables (grouped frequency tables)
histogram
Median and five number summary
Mode
LIS 570
Univariate Analysis
Mason; p. 30
The Information School of the University of Washington
Summary
Interval
Determine whether the distribution is skewed or
symmetrical
Compare median and mean
Use the mean and the standard deviation if the
distribution is not markedly skewed; otherwise
use five number summary (median, extremes,
mid-quartile numbers)
Use the mode in addition if it adds anything
LIS 570
Univariate Analysis
Mason; p. 31
The Information School of the University of Washington
Abstract and Elevator Speech
20-30 second synopsis;
intent: to elicit interest
• Who you are and
what you are doing
• With whom
• Where/How
• Why: What you hope
to find, why the results
may be important
100-300 words; elicit
interest and summarize
• What type of study
• How approached
• When, where
• Why: what you hope
to find, why the results
may be important
LIS 570
Univariate Analysis
Mason; p. 32
The Information School of the University of Washington
Selecting analysis and statistical
techniques*
Specific research question or hypothesis
Determine # of variables in question
Univariate analysis
Bivariate analysis
Multivariate analysis
Determine level of
measurement of variables
Choose univariate
method of analysis
Choose relevant
descriptive statistics
* Source: De Vaus, D.A. (1991) Surveys in Social
Research. Third edition. North Sydney, Australia:
Allen & Unwin Pty Ltd., p133
Choose relevant
inferential statistics
LIS 570
Univariate Analysis
Mason; p. 33
The Information School of the University of Washington
Exercise—sampling distribution
•
•
•
•
Coins, coins!
Probability of head or tails—50%
Each of you is a “sample” for this activity.
Flip the coin 7 times, count the # of times
you get a “head”.
Live demo:
http://www.ruf.rice.edu/~lane/stat_sim/sampling
_dist/index.html
LIS 570
Univariate Analysis
Mason; p. 34
The Information School of the University of Washington
68%
Why is
normality
important?
•
•
•
95%
100%
Use proportions of the normal distribution to determine probabilities associated with any specific
sample.
Sampling Error
Standard Error (SE)—a way for defining and measuring sampling error (exactly, how much error,
on average, should exist between a sample mean and the unknown population mean, simply due
to chance.
LIS 570
Univariate Analysis
Mason; p. 35
The Information School of the University of Washington
Standard Error of the mean
Standard error of the mean (Sm)
Sm =
–
–
–
–
S
S
Standard deviation
Total number in the sample
N
Standard error is inversely related to square root of sample
size
To reduce standard error, increase sample size
Standard error is directly related to standard deviation
When N = 1, standard error is equal to standard deviation
LIS 570
Univariate Analysis
Mason; p. 36
The Information School of the University of Washington
Inferential statistics - univariate analysis
Interval estimates and interval variables
• Estimation of sample mean accuracy—based on
random sampling and probability theory
– Standardize the sample mean to estimate population
mean:
t = sample mean – population mean
estimated SE
– Population mean = sample mean + t * (estimated SE)
LIS 570
Univariate Analysis
Mason; p. 37
The Information School of the University of Washington
Confidence Interval
Utilizes probability theory, assumes normal distribution
• 95% of the samples will fall
within 1 to 2 standard
deviations from the
population mean
• By the same token,
for 95% of samples, the
population mean will be within + or - 2 standard error units
from the sample mean
• E.g., for C.I. 80%, first find the lower and upper t-values that
bind 80% area of the distribution.
• Can state: with 80% confidence interval, the population
mean is: sample mean + t (SE)
LIS 570
Univariate Analysis
Mason; p. 38
The Information School of the University of Washington
Standard Error
(for nominal & ordinal data)
Variable must have only two categories
(may have to combine categories to achieve this)
P = the % in one category of the variable
SB =
PQ
N
Q = the % in the other category of the variable
Total number in the sample
Standard error
for binominal distribution
LIS 570
Univariate Analysis
Mason; p. 39