This document provides an overview of key concepts in statistics including different types of data, measures of central tendency, measures of dispersion, and methods for characterizing diagnostic tests. It discusses nominal, ordinal, interval and ratio data as well as discrete and continuous data. Measures of central tendency covered include mean, median and mode. Measures of dispersion include range, interquartile range, mean deviation, variance, standard deviation and coefficient of variation. The document also defines sensitivity, specificity, positive predictive value, negative predictive value and likelihood ratios for characterizing diagnostic tests. It introduces the receiver operating characteristic curve and how the area under the curve can be used to assess a test's accuracy.

1. A lesson on Statistics:
Data – Types, description and
interpretation
Dr Andrea Josephine R,
2nd year MD PG,
Department of Pediatrics,
ESIC Medical College & PGIMSR, Chennai.

2. Topics
 Types of data
 Measures of central tendency
 Measures of dispersion
 Measures of distribution
 Characterizing diagnostic tests – The test of a test

3. Types of data
1. Nominal:
 Qualitative data
 Characteristics of a variable – Categories
 Mutually exclusive, exhaustive
 No implied order
 E.g. Sex : Male/Female, Demographics
(Urban/Suburban/Rural)

4. Types of data
2. Ordinal:
 Qualitative data – Categories
 Rank/Order into a progression, mutually exclusive,
exhaustive
 Size of the interval not measurable or equal
 E.g. Satisfaction with treatment – Very satisfied /
Somewhat satisfied / Somewhat dissatisfied / Very
dissatisfied

5. Types of data
3. Interval:
 Quantitative data
 Meaningful intervals
 No absolute zero
 Ratio between 2 measurements not meaningful
 E.g. Temperature scale: In degrees Celsius, difference
between 2 measurements quantifiable, but ratio not
meaningful; 0⁰C does not imply a total absence of heat

6. Types of data
4. Ratio:
 Quantitative data
 Absolute zero
 Meaningful ratios
 E.g. Age (years), Weight(kg), Blood pressure(mmHg)

7. Types of data
1. Discrete:
 Only whole numbers possible / distinct categories
 E.g. Number of patients, number of syringes used, Gender,
hair colour
2. Continuous:
 Any value in a continuum
 E.g. Weight, Height, Serum creatinine

8. Measures of central tendency
1. Mean:
 Used for interval & ratio data
 Summation of all values divided by number of values in
the sample
 x = Ʃx
n

9. Measures of central tendency
2. Median:
 Used for ordinal data
 Half of the values lie above it, half below it
 If n is odd, arrange in order: Middle value = median
 If n is even, arrange and take mean of middle 2 values

10. Measures of central tendency
3. Mode:
 Used for nominal data
 Most frequently appearing category
 If 2 categories appear equally, bimodal
 Can be multimodal

11. Measures of dispersion
1. Range:
 Difference between highest and lowest values
 E.g. A set of values 102, 105, 109, 111 and 120. Range is
not 102-120. Range = 120-102 = 18.

12. Measures of dispersion
2. Interquartile range:
 Range of the middle 50% of the data
 Difference between the upper and lower quartile

13. Measures of dispersion
3. Mean deviation:
 Average of the absolute deviations from mean.
 Mean deviation = Ʃ ǀ x – x ǀ
n

14. Example
 Mean Deviation of 3, 6, 6, 7, 8, 11, 15, 16
 Step 1: Find the mean: (3 + 6 + 6 + 7 + 8 + 11 + 15 + 16)/8
= 72/8 = 9
 Step 2: Find the distance of each value from that mean:

15. Example (Contd.)
 Step 3. Find the mean of those distances:
 Mean Deviation = (6 + 3 + 3 + 2 + 1 + 2 + 6 + 7)/8 = 30/8 =
3.75
 So, the mean = 9, and the mean deviation = 3.75
 3.75 away from the middle
 Why take absolute value?

16. Measures of dispersion
4. Variance and Standard deviation:
 Variance(s2) = Mean of the squares of the deviation
= Ʃ (x – x )2
n
 Standard deviation(SD) = Ʃ (x – x)2
√ n
Smaller value of SD Closer the values cluster around
the mean
If a constant is added to all values, mean changes; Variance
and SD remain the same.

17. Measures of dispersion
5. Coefficient of variation(CV):
 CV = SD/Mean
 The units of SD and mean are same, hence CV is an
independent value.
 If both SD and mean are multiplied by a constant, CV
remains the same (Useful in ratio measurements).
 Not useful in interval level data, as CV decreases with
addition of a constant to each value.

18. Skewness
 Refers to the symmetry of the frequency-distribution
curve.
 Value of 0 – Unskewed, Positive value – skewed to the
right, Negative value – skewed to the left.
 Refers to the side of the longer tail, NOT that of the bulk
of the data.

19. Kurtosis
 Refers to the peak of the frequency-distribution curve.
 Mesokurtosis – Normal distribution curve
 Leptokurtosis – Peaked; Platykurtosis - Flat

20. Sensitivity
 Ability of a test to correctly identify patients with disease
 Sensitivity = True positives
True positives + False negatives
Patients
picked up, 80
Undiagnosed
diseased
population, 20
Sensitivity

21. Specificity
 Ability of the test to correctly identify patients who are
disease-free/healthy
 Specificity = True negatives
True negatives + False positives
Healthy,
80
Healthy
mis-
labelled
diseased, 20
Specificity

22. Positive predictive value
 Proportion of patients with positive test results who truly have
disease.
 PPV = True positive
True positive + False positive
 Answers the question: “I have tested
positive. Am I really diseased?”
Truly
diseased
80%
Healthy
mislabelled
diseased
20%
PPV

23. Negative predictive value
 Proportion of patients with negative test results who are
truly disease-free
 NPV = True negatives
True negatives + False negatives
 Answers the question: “I have tested
negative. Am I really disease-free?”
Truly
healthy
80%
Diseased
mis-
labelled
healthy
20%
NPV

24. PPV and NPV
 Highly dependent on the prevalence of a disease in a
given population.
 Less reliable in rare diseases.
 Less transferable from one population to another.

25. Likelihood ratio
 Combines sensitivity and specificity
 Positive likelihood ratio defines the extent to which a
positive test result increases the likelihood of having
disease.
 LR+ = Sensitivity
1 – Specificity
 If the LR + of a test is 1.36, a patient who tests positive is
1.36 times more likely to have the disease than a patient
who tests negative.

26. LR – Interpretation:
 LR+ over 5 - 10: Significantly increases likelihood of the
disease
 LR+ between 0.2 to 5 (esp if close to 1): Does not modify
the likelihood of the disease
 LR+ below 0.1 - 0.2: Significantly decreases the likelihood
of the disease

27. Likelihood ratio
 Negative likelihood ratio defines the extent to which a
negative test result decreases the likelihood of having
disease.
 LR- = 1 – sensitivity
Specificity
 If LR- of a test is 1.5, it means a patient with a negative
test result is 1.5 times more likely to be disease-free than
a patient with a positive test result.

28. LR – Points:
 Independent of disease prevalence
 Specific to the test being used
 Can be applied to the individual patient to evaluate how
worthwhile it is to perform a given test

29. Receiver Operator Characteristics Curve
 If the cut-off for a test is raised, both true and
false positive rate would decrease.
 True positive rate = Sensitivity
 False positive rate = 1 – Specificity.
 A graph between the 2 is ROC curve.

30. ROC curve – Area under curve
 Area under curve – Used to assess overall accuracy of a
test
 Value of 1 – High sensitivity and specificity
 Value of 0.5 – Zero diagnostic capability, Line
of zero discrimination, no better than tossing
a coin.

31. Using ROC curve and AUC to choose
between tests
 ROC curves:

32. References
 Biostatistics: The bare essentials, 3e by Norman and Streiner
 Health services research methods, 2e by Leiyu Shi
 Bewick V, Cheek L, Ball J. Statistics review 13: Receiver operating
characteristic curves. Crit Care. 2004;8(6):508-512.
 AG Lalkhen, A McCluskey. Clinical tests: sensitivity and specificity. Contin Educ
Anaesth Crit Care Pain (2008) 8 (6): 221-223.

33. Thank You