Download frequency distribution

hss2381A – quantitative
methods
Univariate Analysis part 2
Frequency Analysis
WHAT THE HECK ARE ALL THOSE
NUMBERS???
Frequency Distributions
• That’s what a frequency distribution is for—to
help impose order on the data
• A frequency distribution is a systematic
arrangement of data values, with a count of
how many times each value occurred in a
dataset
Uses of Frequency Distributions in Data
Analysis
• First step in understanding your data!
– Begin by looking at the frequency distributions
for all or most variables, to “get a feel” for the
data
– Through inspection of frequency distributions,
you can begin to assess how “clean” the data
are
Data Cleaning
• One aspect of data cleaning involves seeing
whether the frequency distribution contains:
– Outliers: Values that lie outside the normal range
of values, and that may or may not be legitimate
– Wild codes: Impossible or invalid codes, like a
code of “3” for the variable sex when valid codes
are 1 (female) and 2 (male)
Wild Codes
Codes for Sex
Frequency
Percent
1 (Female)
49
49.0%
2 (Male)
47
47.0%
3
1
1.0%
7
2
2.0%
Total
100
100.0%
The codes 3 and 7 are WILD!
Missing Values
•
Frequency
distributions can
help you assess the
pervasiveness of a
thorny problem in
data analysis:
– Missing data
Wanted:
Missing Number!
Description: Data Values
in Important Study
Last seen: Date of
Enrollment
Missing from: My Dataset
If Found: Contact Me!
Inspection for Missing Values
Sex
Frequency
Percent
Valid %
1 (Female)
46
46.0
51.7
2 (Male)
43
43.0
48.3
7 (Refused)
11
11.0
100
100.0
Total
100.0
11.0% of the data are missing because
participants refused to report their sex
Assumptions
•
Frequency distributions can help you
assess validity of certain assumptions for
many statistical tests
– An assumption is a condition presumed to
be true and, when violated, can result in
invalid results
– For many inferential statistics, a normal
distribution (for the dependent variable) is
assumed
Describe Sample
•
Frequency distributions can help you
better understand the type of people
who are in your study sample:
– What percent are men?
– What percent are African American?
– What percent have a college degree?
Answer Descriptive Questions
•
Frequency distributions can sometimes
be used to answer descriptive research
questions
•
BUT…inferential statistics are almost
always needed, because they allow you
to draw inferences about a broader
group than the study sample
Frequency Distributions in SPSS
•
•
Use the Analyze  Descriptive Statistics
 Frequencies command
Click “Analyze” in the top toolbar menu,
which brings up a pop-up menu; select
Descriptives
Frequencies Command in SPSS
•
•
•
All variables in
dataset are listed in
box on left
Use arrow to move
desired variable
into slot marked
“Variable(s)”
Pushbuttons
provide various
options
Frequencies:
Statistics Options in SPSS
•
•
Many available
options within
Frequencies:
Statistics
Here we see that
we can select
statistics for
skewness and
kurtosis
Frequencies:
Chart Options in SPSS
•
•
•
The Charts option allows
you to create bar charts,
pie charts, and histograms
Normal curve
superimposed: An option
for Histograms
Chart values can be
Frequencies or Percentage
(not available for
Histograms)
Graphs in SPSS
•
An even wider array of graphs can be
created using the Graphs menu on the
main toolbar
Characteristics of a Data Distribution
• Shape (Chapter 2)
• Central tendency
• Variability
– Both central tendency and variability can be
expressed by indexes that are descriptive
statistics
Central Tendency
• Indexes of central tendency provide a single
number to characterize a distribution
• Measures of central tendency come from
the center of the distribution of data values,
indicating what is “typical,” and where data
values tend to cluster
• Popularly called an “average”
Central Tendency Indexes
• Three alternative indexes:
– The mode
– The median
– The mean
The Mode
• The mode is the
score value with the
highest frequency;
the most “popular”
score
– Age: 26 27 27 28
29 30 31
– Mode = 27
2.5
2.0
1.5
1.0
Std. D
.5
Mean
N = 7
0.0
26.0
27.0
28.0
29.0
AGE

The mode
30.0
31.0
The Mode: Advantages
• Can be used with data measured on any
measurement level (including nominal level)
• Easy to “compute”
• Reflects an actual value in the distribution,
so it is easy to understand
• Useful when there are 2+ “popular” scores
(i.e., in multimodal distributions)
The Mode: Disadvantages
• Ignores most information in the distribution
• Tends to be unstable (i.e., value varies a lot
from one sample to the next)
• Some distributions may not have a mode (e.g.,
10, 10, 11, 11, 12, 12)
The Median
• The median is the
score that divides the
distribution into two
equal halves
• 50% are below the
median, 50% above
– Age: 26 27 27 28 29
30 31
– Median (Mdn) = 28
2.5
2.0
1.5
1.0
Std. De
.5
Mean =
N = 7.0
0.0
26.0
AGE
27.0
28.0
29.0
30.0

The median
31.0
The Median: Advantages
• Not influenced by outliers
• Particularly good index of what is “typical”
when distribution is skewed
• Easy to “compute”
• Appropriate when data are ordinal level
The Median: Disadvantages
• Does not take actual data values into
account—only an index of position
• Value of median not necessarily an actual
data value, so it is more difficult to
understand than mode
The Mean
• The mean is the
arithmetic average
2.5
2.0
• Data values are
summed and divided
by N
1.5
1.0
Std. Dev =
.5
Mean = 2
N = 7.00
0.0
– Age: 26 27 27 28 29
30 31
– Mean = 28.3
26.0
27.0
28.0
29.0
AGE

The mean
30.0
31.0
The Mean (cont’d)
• Most frequently used measure of central
tendency—usually preferred for interval- and
ratio-level data
• Equation:
M = ΣX ÷ N
• Where:
M = sample mean
Σ = the sum of
X = actual data values
N = number of people
The Mean: Advantages
• The balance point in the distribution:
– Sum of deviations above the mean always
exactly balances those below it
• Does not ignore any information
• The most stable index of central tendency
• Many inferential statistics are based on the
mean
The Mean: Disadvantages
• Sensitive to outliers
• Gives a distorted view of what is “typical”
when data are skewed
• Value of mean is often not an actual data
value
The Mean: Symbols
• Sample means:
– In reports, usually symbolized as M
– In statistical formulas, usually symbolized as
x(pronounced X bar)
• Population means:
– The Greek letter μ (mu)
Central Tendency in Normal
Distributions
• In a normal
distribution, all
three indexes
coincide
Central Tendency in Skewed
Distributions
• In a skewed distribution, the mean is pulled
“off center” in the direction of the skew
Variability
• Variability concerns how spread out or
dispersed data values in a distribution are
• Two distributions with the same mean could
have different dispersion
Variability (cont’d)
• High variability: A
heterogeneous
distribution (A)
• Low variability: A
homogeneous
distribution (B)
Indexes of Variability
• Range
• Interquartile range
• Standard deviation
• Variance
The Range
• Range: The difference between the highest
and lowest value in the distribution
• Weights (pounds):
110 120 130 140 150 150 160 170 180 190
• The range here is 80 (190 – 110)
The Range: Advantages
• Easy to compute
• Readily understood
• Communicates information of interest to
readers of a report
The Range: Disadvantages
• Depends on only two scores, does not take all
information into account
• Sensitive to outliers
• Tends to be unstable—fluctuates from sample
to sample
• Influenced by sample size
The Interquartile Range
• Interquartile range (IQR): Based on quartiles
– Lower quartile (Q1): Point below which 25% of scores
lie
– Upper quartile (Q3): Point below which 75% of scores
lie
• IQR = Q3 - Q1
– IQR is the range of scores within which the middle
50% of scores lie
Consider this dataset (yanked from Wikipedia)
Notice that Q2 is always the median
N=11
n+1 = 12
Q2 = median = entry # (n+1)/2
Q1 = upper = entry # (n+1)/4
Q3 = lower = entry # 3(n+1)/4
Q1 = 3rd entry = 105
Q3 = 9th entry = 115
IQR = Q3-Q1 = 115-105 = 10
The Interquartile Range (cont’d)
• Another Example: Weights (pounds):
110 120 130 140 150 160 170 180 190
• The IQR is 50.0 (175 – 125)
• Let’s see how we get that….
Number of entry
Value
1
110
2
120
3
130
4
140
5
150
6
160
7
170
8
180
9
190
Step 1 = where is the median?
Quartile
Number of entry
Value
1
110
2
120
3
130
4
140
5
150
6
160
7
170
8
180
9
190
Quartile
Q1=125
Q2 = median
Q1 will be entry # (9+1)/4 = 2.5 = halfway between 120 and 130
Q3 will be entry # 3(9+1)/4 = 7.5 = halfway between 170 and 180
Q3=175
What if we have an even number?
• IQR Example: Weights (pounds):
110 120 130 140 150 150 160 170 180 190
• The IQR is 45.0 (172.5 – 127.5)
• Let’s see how we get that…
Number of entry
Value
1
110
2
120
3
130
4
140
5
150
6
150
7
160
8
170
9
180
10
190
Step 1 = where is the median?
Quartile
Number of entry
Value
1
110
2
120
3
130
4
140
5
150
6
150
7
160
8
170
9
180
10
190
Q1=127.5
Q2=Median = 150
Q1 will be entry # (10+1)/4 = 2.75 = ¾ of the way between 120 and 130
Or... 120 + [(130-120) x 0.75] = 127.5
Number of entry
Value
1
110
2
120
3
130
4
140
5
150
6
150
7
160
8
170
9
180
10
190
Q1=127.5
Q2=Median = 150
Q1=172.5
Q3 will be entry # 3(10+1)/4 = entry # 8¼ or 25% of the distance between 170 & 180
Or... 170 + [(180-170) x 0.25] = 172.5
Number of entry
Value
1
110
2
120
3
130
4
140
5
150
6
150
7
160
8
170
9
180
10
190
IQR = q3-q1 = 172.5 – 127.5 = 45.0
Q1=127.5
Q2=Median = 150
Q1=172.5
If you want to check your work, use any stats software,
or an online IQR calculator, such as:
http://www.alcula.com/calculators/statistics/interquartile-range/
The Interquartile Range: Advantages
• Reduces influence of outliers and extreme
scores in expressing variability
• Uses more information than the range
• Important in evaluating outliers
• Appropriate as index of variability with
ordinal measures
The Interquartile Range: Advantages
The closer the clustering of
values around the median,
the smaller the interquartile
range
Small IQR shows
clustering around the
median.
Why is this useful?
The Interquartile Range:
Disadvantages
• Is not particularly easy to compute
• Is not well understood
• Does not take all values into account
The Standard Deviation
• Standard deviation (SD): An index that conveys
how much, on average, scores in a distribution
vary
• SDs are based on deviation scores (x),
calculated by subtracting the mean from each
person’s original score
x=X-M
Standard Deviation Interpretation
• In a normal distribution, a fixed percentage
of cases lie within certain distances from the
mean:
We will do more with SD and
variance...
Measurement Scales and Descriptive
Statistics
Scale
Central
Variability
Tendency Index Index
Nominal
Mode
--
Ordinal
Median
Range, IQR
Interval and
ratio
Mean
Standard
deviation,
Variance
Uses of Descriptive Statistics
• Indexes of central tendency and variability
are used to:
– Understand data, get a “big picture”
– Evaluate outliers and need for strategies to
address problems (e.g., using a trimmed mean
that recalculates mean after deleting a fixed
percentage (e.g., 5% from either end)
– Describe research participants (e.g., their age,
education, length of illness)
– Answer descriptive questions
Descriptive Statistics in SPSS
• Can be obtained through Analyze 
Descriptive Statistics and are obtained in
three programs within that broad umbrella
(each has slightly different options):
– Frequencies  Statistics
– Descriptives  Options
– Explore  Statistics
Descriptive Statistics in SPSS
Frequencies
•
•
•
•
Percentile values
Central tendency
Dispersion (variability)
Skewness and Kurtosis
Descriptive Statistics in SPSS
Descriptives
•
•
•
•
•
Mean (no median)
Dispersion (variability)
Skewness and Kurtosis
No percentiles
BUT has good display options
Example
• We ask a class of 10 students what their
weight in pounds is. We get:
Student
Weight
1
2
3
4
5
6
7
8
9
10
98
102
175
165
160
148
320
102
111
55
Step 1 – rank the data
Student
Weight
Student
Weight
1
2
3
4
5
6
7
8
9
10
98
102
175
165
160
148
320
102
111
55
10
1
2
8
9
6
5
4
3
7
55
98
102
102
111
148
160
165
175
320
Total = 1436
Student
Weight
10
1
2
8
9
6
5
4
3
7
55
98
102
102
111
148
160
165
175
320
Total = 1436
Mean = total/number of students
= 1436/10
= 143.6
Mode = most common response
= 102
Student
Weight
10
1
2
8
9
6
5
4
3
7
55
98
102
102
111
148
160
165
175
320
Total = 1436
How do we find the median?
Student
Weight
10
1
2
8
9
6
5
4
3
7
55
98
102
102
111
148
160
165
175
320
Total = 1436
How do we find the median?
Find the middle value. But
since there are 10 values total,
there are 2 middle values
Then find the midpoint between
the two by computing the mean
of those two:
(111+148)/2 = 129.5
Student
Weight
10
1
2
8
9
6
5
4
3
7
55
98
102
102
111
148
160
165
175
320
Total = 1436
How do we find the range?
Find maximum:
Find minimum
Subtract them:
320
55
265
How do we find the IQR?
Student
Weight
10
1
2
8
9
6
5
4
3
7
55
98
102
102
111
148
160
165
175
320
<- Q1 = 101.0
<-Q2=median = 129.5
<- Q3 = 167.5
Total = 1436
IQR = Q3-Q1 = 167.5 – 101.0 = 66.5
Homework
• P. 57 A1, A2, A3