Statistic in Health Care Management Week 2
gregueira82
chapter4.pdf
Chapter 4
Summarizing Data Collected in
the Sample
Learning Objectives
• Distinguish between dichotomous, ordinal,
categorical, and dichotomous variables
• Identify appropriate numerical and graphical
summaries for each variable type
• Compute a mean, median, standard deviation,
quartiles and range for a continuous variable
Learning Objectives
• Construct a frequency distribution table for
dichotomous, categorical and ordinal variables
• Provide an example of when the mean is a
better measure of location than the median
• Interpret the standard deviation of a continuous
variable
Learning Objectives
• Generate and interpret a box plot for a
continuous variable
• Produce and interpret side-by-side box plots
• Differentiate between a histogram and a bar
chart
Variable Types
• Dichotomous variables have 2 possible responses
(e.g., Yes/No)
• Ordinal and categorical variables have more than two
responses and responses are ordered and unordered,
respectively
• Continuous (or measurement) variables assume in
theory any values between a theoretical minimum and
maximum
Biostatistics
Two Areas of Applied Biostatistics:
Descriptive Statistics
– Summarize a sample selected from a population
Inferential Statistics
– Make inferences about population parameters based on sample statistics.
Vocabulary
• Data elements/data points
• Subjects/units of measurement
• Population Vs. Sample
Sample vs Population
• Any summary measure computed on a sample
is a statistic
• Any summary measure computed on a
population is a parameter
n = sample size
N = population size
Example 4.1.
Dichotomous Variable
Frequency Distribution Table
Hypertension Treatment
Frequency Relative Frequency (%)
No 2313 65.5%
Yes 1219 34.5%
3532 100.0%
Relative Frequency Bar Chart for
Dichotomous Variable
Categorical Outcome
Sample: n=50
Population: Patients at health center
Variable: Marital status
Marital Status Number of Patients
Married 24
Separated 5
Divorced 8
Widowed 2
Never Married 11
Total 50
Categorical Outcome
Frequency Distribution Table
Marital Status Number of
Patients (f)
Relative Frequency (f/n)
Married 24 0.48
Separated 5 0.10
Divorced 8 0.16
Widowed 2 0.04
Never Married 11 0.22
Total 50 1.00
Frequency Bar Chart
Ordinal Outcome
Sample: n=50
Population: Patients at health center
Variable: Self-reported current health status
Health Status Number of Patients
Excellent 19
Very Good 12
Good 9
Fair 6
Poor 4
Total 50
Ordinal Outcome
Frequency Distribution Table
Heath Status Freq. Rel. Freq. Cumulative Freq
Cumulative Rel. Freq.
Excellent 19 38% 19 38%
Very Good 12 24% 31 62%
Good 9 18% 40 80%
Fair 6 12% 46 92%
Poor 4 8% 50 100%
50 100%
Relative Frequency Histogram
0
5
10
15
20
25
30
35
40
Poor Fair Good Very Good Excellent
Health Status
%
Example 4.2.
Ordinal Variable
Frequency Distribution Table
Blood Pressure Categories
Frequency Relative Frequency (%)
Normal 1206 34.1%
Pre-hypertension 1452 41.1%
Stage I hypertension 653 18.5%
Stage II hypertension 222 6.3%
Total 3533 100.0%
Relative Frequency Histogram for Ordinal
Variable
Continuous Variables
• Assume, in theory, any value between a
theoretical minimum and maximum
• Quantitative, measurement variables
Continuous Variable
• Population: Patients 50 years of age with
coronary artery disease
• Sample: n = 7 patients
• Outcome: Systolic blood pressure (mmHg)
Continuous Variable
Sample data
X
100
110
114
121
130
130
160
Continuous Variable
6.123 7
865
n
X X
X
100
110
114
121
130
130
160
865
n
X X mean Sample
Continuous Variable
Consider a second sample from the same population.
We record SBP on each subject in the second sample:
120 121 122 124 125 126 127
n = 7
= 865 / 7 = 123.6.
What is different between the 2 samples? X
Continuous Variable
• Dispersion
X (X- )
100 -23.6
110 -13.6
114 -9.6
121 -2.6
130 6.4
130 6.4
160 36.4
865 0
X
Continuous Variable
• Dispersion
X (X- )
100 -23.6
110 -13.6
114 -9.6
121 -2.6
130 6.4
130 6.4
160 36.4
865 0
X Mean Absolute Deviation (MAD):
n
| X - X| Σ = MAD
Continuous Variable
X X
1n
)XΣ(X s
2 2
374.6 6
2247.72 s
2
Sample Variance:
X (X- ) (X- )2
100 -23.6 556.96
110 -13.6 184.96
114 -9.6 92.16
121 -2.6 6.76
130 6.4 40.96
130 6.4 40.96
160 36.4 1324.96
865 0 2247.72
Continuous Variable
• Sample Standard Deviation:
s = s 2
4.196.374 s
Standard Summary: n=7, X = 123.6, s=19.4
Median
Median
100 110 114 121 130 130 160
Median holds 50% of values above and 50% of values below
Order data For n odd – median is middle value For n even – median is mean of 2
middle values
Quartiles
Q1 = first quartile holds approximately 25% of the
scores at or below it and
Q3 = third quartile holds approx. 25% of the
scores at or above it
Q2 = ??
Continuous Variable
Median
Order data
100 110 114 121 130 130 160
Q1 Q3
Box and Whisker Plot
100 110 120 130 140 150 160
Min Q1 Median Q3 Max
Comparing Samples with
Box and Whisker Plots
100 110 120 130 140 150 160
Summarizing Location and Variability
• When there are no outliers, the sample mean
and standard deviation summarize location
and variability
• When there are outliers, the median and
interquartile range (IQR) summarize location
and variability, where IQR = Q3-Q1
Example
Sample: n=51 participants in a study of
cardiovascular risk factors.
Variable: age (years)
60 62 63 64 64 65 65 65 65 65 65
66 66 66 66 66 67 67 67 68 68 68
70 70 70 71 71 72 72 73 73 73 73
73 73 75 75 75 76 76 77 77 77 77
77 79 82 83 85 85 87
Example
Sample mean: 71.3 =
51
3637 =
n
XΣ = X
Sample variance:
41.4 = 50
/51)(3637 - 261,439 =
1 -n
/n)X(Σ - XΣ = s
222
2
Sample standard deviation:
6.4 = 41.4 = s
Standard Summary: n=51, X = 71.3, s=6.4
Outliers
IQR = Interquartile Range = Q3 - Q1
= range of middle half of the data
Outliers are values which either:
exceed Q3 + 1.5 IQR, or
fall below Q1 - 1.5 IQR
Or outliers are outside + 3s X
Check for Outliers in Example
• Q1=66, Q3=76, IQR=10
– Lower=66-1.5(10)=51
– Upper=76+1.5(10)=91
• + 3s = 52.1 to 90.5X
Presenting Data
• Suppose we collapse ages into 5 mutually exclusive and
exhaustive categories:
Age Class Number of Individuals (freq.)
60-64 5
65-69 17
70-74 12
75-79 12
80-84 2
85-89 3
Presenting Data
Cumulative
Age Class Freq Rel Freq Freq Rel Freq
60-64 5 0.10 5 0.10
65-69 17 0.33 22 0.43
70-74 12 0.24 34 0.67
75-79 12 0.24 46 0.91
80-84 2 0.04 48 0.95
85-89 3 0.06 51 1.00
Total 51 1.00
Frequency Histogram
0 2 4 6 8
10 12 14 16 18
60-
64
65-
69
70-
74
75-
79
80-
84
85-
89
Age Class
F r e q
u e n
c y
Example 4.3.
Summarizing Continuous Variables
Diastolic blood pressures in n=10 randomly
selected participants attending the seventh
examination of the Framingham Offspring
Study
76 64 62 81 70
72 81 63 67 77
Summarizing Location
• What is a typical diastolic blood pressure?
Sample Mean
= Sum of diastolic blood pressures/n
= 713/10 = 71.3
Notation
• Let X represent the outcome of interest (e.g.,
X=diastolic blood pressure)
n
X X mean Sample
Summarizing Variability
• Sample range
= maximum–minimum=81–62 = 19
• Sample variance
1n
)x(x s
2
2
Sample Variance
DBP Deviation from Mean
76 (76 - 71.3) = 4.7
64 (64 - 71.3) = -7.3
62 (62 - 71.3) = -9.3
81 9.7
70 -1.3
72 0.7
81 9.7
63 -8.3
67 -4.3
77 5.7
S X = 71.3 S Deviations from Mean = 0
Sample Variance
DBP Deviation from Mean Squared Deviations
76 (76 - 71.3) = 4.7 22.09
64 (64 - 71.3) = -7.3 53.29
62 (62 - 71.3) = -9.3 86.49
81 9.7 94.09
70 -1.3 1.69
72 0.7 0.49
81 9.7 94.09
63 -8.3 68.89
67 -4.3 18.49
77 5.7 32.49
S X = 71.3 S Deviations = 0 S Deviations2 = 472.10
Sample Variance and Sample Standard
Deviation
46.52 9
10.472
1n
)x(x s
2
2
2.746.52 1n
)x(x s
2
Median
• Median holds 50% of values above and 50%
of values below
– Order data
– For n odd – median is middle value
– For n even – median is mean of 2 middle values
Median = 71
62 63 64 64 70 | 72 76 77 81 81
Quartiles
• Q1 = first quartile = holds 25% of values
below it
• Q3 = third quartile = holds 25% of values
above it
Median = 71
62 63 64 64 70 | 72 76 77 81 81
Q1 Q3
Determining Outliers
• Outliers are values
below Q1-1.5(Q3-Q1) or
above Q3+1.5(Q3-Q1)
• In Example 4.3,
lower limit = 64-1.5(77-64) = 44.5
and upper limit=77+1.5(77-64) = 96.5
• Outliers?
• Mean or Median? s or IQR?
Box Plot for Continuous Variable
60
65
70
75
80
d b p
Numerical and Graphical Summaries
• Dichotomous and categorical
– Frequencies and relative frequencies
– Bar charts (freq. or relative freq.)
• Ordinal
– Frequencies, relative frequencies, cumulative frequencies and cumulative relative frequencies
– Histograms (freq. or relative freq.)
Numerical and Graphical Summaries
• Continuous
– Mean, standard deviation, minimum, maximum, range, median, quartiles, interquartile range
– Box plot
