Updated: 10/3/2020

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Descriptive Statistics
  • Mean
    • average of all observation
    • mean = (sum of all observations)/(sample size)
  • Median 
    • the middle value of all observations  
    • if sample size is odd
      • median = ((n+1)/2)th largest value
    • if the sample size is even
      • median = the average of the (n/2)th and ((n/2)+1)th largest value 
  • Mode
    • the most commonly occurring value
    • if there is more than 1 most commonly occurring value, there are as many modes as most commonly occurring values   
    • in decreasing order of resistance to outliers, mode > median > mean 
Types of Distributions
  • Normal
    • aka Gaussian, bell-shaped
    • for continuous variables
    • mean = median = mode
  • Bi-modal
    • distribution has 2 humps (each being a relative mode)
    • if symmetrical, mean = median
  • Skewed  
    • positive skew
      • asymmetrical with tail trailing off to right
      • mean > median > mode  
    • negative skew
      • asymmetrical with tail trailing off to left
      • mean < median < mode
    • mean very sensitive to skew 
    • median somewhat resistant to skew
    • mode very resistant to skew
  • Other
    • non-continuous variable types have their own distributions 
      • e.g., binary, categorical, ordinal, binomial, and count variables 
Characteristics of the Normal Distribution
  • For continuous variables
  • Defined entirely by 2 parameters
  • Mean (µ)
    • standard deviation (σ)
  • A certain percentage of all observations will always fall within +/- certain standard deviations of the mean  
    • +/- 1 standard deviation = 68%  
    • +/- 2 standard deviations = 95%    
    • +/- 3 standard deviations = 99.7%
 Regression to the Mean
  • Phenomenon in which sample points which were initially extreme often become closer to the mean in future measurements
  • Most points will fall near on the average; therefore, extreme points are often a result of "luck" (e.g., a student performs particularly poor on an exam but normally performs at the average level)
  • Has significance for study design
    • e.g., patients with high blood pressure may improve after taking an experimental anti-hypertensive, but that improvement on the next measurement may be due to regression to the mean rather than the treatmentt
      • the solution is to compare a control and experimental group. 
Measures of Variability
  • Standard deviation
    • a statistical measure that demonstrates how close together or spread apart the data is 
      • if data is closer together, the standard deviation will be smaller (and vice versa)
    • often designated by σ
    • equation
      • square root[(sum of the differences between each data point and the mean squared)/n]
  • Standard error
    • a statistical measure that demonstrates how far the sample mean is from the true population mean
      • helps determine confidence intervals
    • equation
      • standard deviation/square root of n
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Questions (15)
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(M1.ST.15.19) A study on cholesterol levels of a town in rural Idaho is performed, of which there are 1000 participants. It is determined that in this population, the mean LDL is 200 mg/dL, with a standard deviation of 50 mg/dL. If the population has a normal distribution, how many people have a cholesterol less than 300 mg/dL? Tested Concept

QID: 102598
1

680

8%

(3/38)

2

925

3%

(1/38)

3

950

24%

(9/38)

4

975

58%

(22/38)

5

997

3%

(1/38)

M 1 D

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(M1.ST.14.17) On morning labs, a patient's potassium comes back at 5.9 mEq/L. The attending thinks that this result is spurious, and asks the team to repeat the electrolytes. Inadvertently, the medical student, intern, and resident all repeat the electrolytes that same morning. The following values are reported: 4.3 mEq/L, 4.2 mEq/L, and 4.2 mEq/L. What is the median potassium value for that patient that day including the first value? Tested Concept

QID: 103979
1

4.2 mEq/L

13%

(5/38)

2

4.25 mEq/L

71%

(27/38)

3

4.3 mEq/L

3%

(1/38)

4

4.65 mEq/L

5%

(2/38)

5

1.7 mEq/L

5%

(2/38)

M 1 E

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(M1.ST.13.67) A clinical trial investigating a new biomedical device used to correct congenital talipes equinovarus (club foot) in infants has recently been published. The study was a preliminary investigation of a new device and as such the sample size is only 20 participants. The results indicate that the new biomedical device is less efficacious than the current standard of care of serial casting (p < 0.001), but the authors mention in the conclusion that it may be due to a single outlier--a patient whose foot remained uncorrected by the conclusion of the study. Which of the following descriptive statistics is the least sensitive to outliers? Tested Concept

QID: 103884
1

Mean

0%

(0/35)

2

Median

26%

(9/35)

3

Mode

49%

(17/35)

4

Standard deviation

17%

(6/35)

5

Variance

3%

(1/35)

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(M1.ST.13.69) A case series is published on the duration of symptoms in children presenting to their primary medical doctor with suspected viral upper respiratory infection. Figure A is presented in the article. What is mean, median, and mode of the duration of symptoms in this case series? Tested Concept

QID: 103886
FIGURES:
1

Mean 8, median 7, mode 9

47%

(16/34)

2

Mean 7, median 5, mode 13

9%

(3/34)

3

Mean 8, median 8, mode 9

24%

(8/34)

4

Mean 7, median 8, mode 9

6%

(2/34)

5

Not enough information provided.

6%

(2/34)

M 1 E

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(M1.ST.13.66) You are conducting a study on hypertension for which you have recruited 60 African-American adults. If the biostatistician for your study informs you that the sample population of your study is approximately normal, the mean systolic blood pressure is 140 mmHg, and the standard deviation is 7 mmHg, how many participants would you expect to have a systolic blood pressure between 126 and 154 mmHg? Tested Concept

QID: 103883
1

10 participants

5%

(11/202)

2

41 participants

17%

(35/202)

3

57 participants

61%

(124/202)

4

68 participants

4%

(9/202)

5

Not enough information provided.

8%

(16/202)

M 1 E

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