Provide reflection of this week that contains 1 paragraph, which address at least one or two of the following topics
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Biostatistics_CH04_SummarizingData1-2.pptxChapter 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
| Marital Status | Number of Patients |
| Married | 24 |
| Separated | 5 |
| Divorced | 8 |
| Widowed | 2 |
| Never married | 11 |
| Total | 50 |
Categorical Outcome (1 of 2)
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
| Health Status | Number of Patients |
| Excellent | 19 |
| Very good | 12 |
| Good | 9 |
| Fair | 6 |
| Poor | 4 |
| Total | 50 |
Ordinal Outcome (1 of 2)
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
Health Status
%
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)
23
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 |
Continuous Variable (6 of 9)
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 |
Mean absolute
deviation (MAD):
Continuous Variable (7 of 9)
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 (8 of 9)
Continuous Variable (9 of 9)
Sample standard deviation
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 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
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.
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
Age Class
Frequency
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
