Healthcare Statistics

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Chapter 4

Summarizing Data Collected in the Sample

Learning Objectives (1 of 3)

Distinguish between dichotomous, ordinal, categorical, and continuous 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 (2 of 3)

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 (3 of 3)

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 two 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 versus 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

Relative Frequency Bar Chart for Dichotomous Variable

Sample: n = 50

Population: Patients at health center

Variable: Marital status

Categorical Outcome (1 of 2)

Marital Status	Number of Patients
Married	24
Separated	5
Divorced	8
Widowed	2
Never married	11
Total	50

Categorical Outcome (2 of 2)

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

Sample: n =50

Population: Patients at health center

Variable: Self-reported current health status

Ordinal Outcome (1 of 2)

Health Status	Number of Patients
Excellent	19
Very good	12
Good	9
Fair	6
Poor	4
Total	50

Ordinal Outcome (2 of 2)

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

Example 4.2.
Ordinal Variable

Frequency Distribution Table

Relative Frequency Histogram
for Ordinal Variable

Assume, in theory, any value between a theoretical minimum and maximum
Quantitative, measurement variables

Continuous Variable (1 of 9)

Population: Patients 50 years of age with coronary artery disease
Sample: n = 7 patients
Outcome: Systolic blood pressure (mmHg)

Continuous Variable (2 of 9)

Sample data

X 100 110 114 121 130

130 160

Continuous Variable (3 of 9)

X 100 110 114 121 130

130 160

865

Continuous Variable (4 of 9)

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 two samples?

Continuous Variable (5 of 9)

Dispersion

Continuous Variable (6 of 9)

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

Dispersion

Mean absolute

deviation (MAD):

Continuous Variable (7 of 9)

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

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

160 36.4 1324.96

865 0 2247.72

Continuous Variable (8 of 9)

Continuous Variable (9 of 9)

Sample standard deviation

Standard summary

n = 7, X = 123.6, s = 19.4

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 two middle values

Quartiles

Q1 = first quartile holds approximately 25% of the scores at or below it.
Q3 = third quartile holds approximately 25% of the scores at or above it.
Q2 = ??

Continuous Variable

Median

Order data

100 110 114 121 130 130 160

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.

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

79 82 83 85 85 87

Example (1 of 2)

Example (2 of 2)

Sample mean:

Sample variance:

Sample standard deviation:

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 that either:
Exceed Q3 + 1.5 IQR
Fall below Q1 – 1.5 IQR
Or, are outside ± 3s

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 90.5

Presenting Data (1 of 2)

Suppose we collapse ages into five 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 (2 of 2)

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

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)

Summarizing Variability

Sample range:

= maximum – minimum = 81 – 62 = 19

Sample variance:

Sample Variance (1 of 2)

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 (2 of 2)

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

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 two 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—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

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 (1 of 2)

Numerical and Graphical
Summaries (2 of 2)

Continuous
Mean, standard deviation, minimum, maximum, range, median, quartiles, interquartile range
Box plot

PoorFairGoodVery GoodExcellent

Health Status

123

865

mean

Sample

MAD

)

Σ(X

374.6

2247.72

374

71.3

3637

41.4

/51

)

(3637

261,439

)

(

6.4

41.4

60-

65-

70-

75-

80-

85-

Age Class

Frequency

)

472

)

dbp