1. Descriptive and inferential statistics. Confidence interval
Georgi Iskrov, MBA, MPH, PhD
Department of Social Medicine

2. Outline Descriptive statistics Measures of central tendency
Measures of spread
Normal distribution
Central limit theorem
Outliers
Inferential statistics
Confidence interval

3. Population vs Sample
Population includes all objects of interest, whereas sample is only a portion of the population.
Parameters are associated with populations and statistics with samples.
Parameters are usually denoted using Greek letters (μ, σ), while statistics are usually denoted using Roman letters (x, s).

4. Descriptive vs Inferential statistics
We compute statistics and use them to estimate parameters.
The computation is the first part of the statistical analysis (Descriptive Statistics) and the estimation is the second part (Inferential Statistics).
Descriptive Statistics
The procedure used to organise and summarise masses of data.
Inferential Statistics
The methods used to find out something about a population, based on a sample.

5. Inferential statistics
Population
Parameters
Sampling
From population to sample
Sample
Statistics
From sample to population
Inferential statistics

6. Descriptive statistics
Organising data
Tables
Graphs
Summarising data
Central tendency (location)
Variation (spread)

7. Descriptive statistics
Organising data
Tables
Frequency distributions
Relative frequency distributions
Graphs
Bar chart
Histogram
Box plot

8. Frequency distribution
Frequency distribution of survival for both groups
Survival Frequency
14 2
17 1
21 1
22 2
23 1
24 2
25 1
27 1
28 1
29 1
31 1
33 1
34 2
35 1
39 1
41 1
Total 20
Experimental group (10 patients)
Individual survival in months:
23 27
17 34
41 28
22 33
29 14
Classes of values
Control group (10 patients)
Individual survival in months:
24 31
39 35
34 24
14 21
25 22

9. Relative frequency distribution
Relative frequency distribution of survival for both groups
Survival Frequency Percent Cumulative percent
% 10%
% 15%
% 20%
% 30%
% 35%
% 45%
% 50%
% 55%
% 60%
% 65%
% 70%
% 75%
% 85%
% 90%
% 95%
% 100%
Total % 9

10. Grouped relative frequency distribution
Relative frequency distribution of survival for both groups
Survival Frequency Percent Cumulative Percent
10 – % 10%
15 – % 15%
20 – % 45%
25 – % 65%
30 – % 85%
35 – % 95%
40 – % 100%
Total %
Classes of intervals
What rules to follow when groupping data?

11. Descriptive statistics
Summarising data:
Central tendency (or sample’s middle value)
Mean
Median
Mode
Spread (or summary of differences within groups)
Range
Interquartile range
Variance
Standard deviation

12. Mean Most commonly called average. Experimental group (10 patients)
Individual survival in months:
23 27
17 34
41 28
22 33
29 14
Experimental group (10 patients)
Individual survival in months:
23 27
17 34
41 28
22 33
29 14
Control group (10 patients)
Individual survival in months:
14 21
25 22

13. Mean Mean is the balance point.
Means can be heavily affected by outliers (data points with extreme values unlike the rest).
Outliers can make the mean a bad measure of central tendency or common experience.

14. Median The middle value when a variable’s values are ranked in order.
The point that divides a distribution into two equal halves.
When data are listed in order, the median is the point at which 50% of the cases are above and 50% below it.
The 50th percentile.

15. Median Control group (10 patients) Individual survival in months: 1421 22 24 25 31 34 35 39
Median = 24.5
(five cases above, five below)

16. Median
The median is unaffected by outliers, making it a better measure of central tendency, better describing the “typical person” than the mean when data are skewed.
If the recorded values for a variable form a symmetric distribution, the median and mean are identical.
In skewed data, the mean lies further toward the skew than the median.

17. Mode The most common data point is called the mode.
Individual survival data for the control group are:
14, 21, 22, 24, 24, 25, 31, 34, 35, 39
It is possible to have more than one mode.
If all values are unique, there is no mode.
Mode may mot be at the center of a distribution.

18. Mode
It may give you the most likely experience rather than the typical or central experience.
In symmetric distributions, the mean, median and mode are the same.
In skewed data, the mean and median lie further toward the skew than the mode.
Skewed
Symmetric
Mean
Median
Mode
Mode
Median
Mean

19. Spread Variation of the recorded values on a variable.
The larger the spread, the further the individual cases are from the mean.
The smaller the spread, the closer the individual scores are to the mean.
Mean
Mean

20. Range
The spread, or the distance, between the lowest and highest values of a variable.
To get the range for a variable, you subtract its lowest value from its highest value.
Experimental group (10 patients)
Individual survival in months:
23 27
17 34
41 28
22 33
29 14
Range = 41 – 14 = 27
Control group (10 patients)
Individual survival in months:
24 31
39 35
34 24
14 21
25 22
Range = 39 – 14 = 25

21. Standard deviation
Standard deviation takes into account all individual deviations.
A deviation is the distance away from the mean of a case’s score.
Experimental group’s SD = 8.13 months
Control group’s SD = 7.64 months

22. Standard deviation
The larger standard deviation, the greater amounts of variation around the mean.
Standard deviation is equal to 0, only when all values are the same.
Like the mean, the standard deviation will be inflated by an outlier case value.

23. Interquartile range
The interquartile range (IQR) is a measure of variability, based on dividing a data set into quartiles.
Quartiles divide a rank-ordered data set into four equal parts. The values that divide each part are called the first, second, and third quartiles; and they are denoted by Q1, Q2, and Q3, respectively.
IQR is equal to Q3 minus Q1.

24. Central tendency and spread
Central tendency: Mean, mode and median
Spread: Range, interquartile range, standard deviation
Mistakes:
Focusing on only the mean and ignoring the variability
Standard deviation and standard error of the mean
Variation and variance
What is best to use in different scenarios?
Symmetrical data: mean and standard deviation
Skewed data: median and interquartile range

25. Important rules
When a constant is added to every observation, the new sample mean is equal to original mean plus the constant.
When a constant is added to every observation, the standard deviation is unaffected.
When every observation is multiplied by the same constant, the new sample mean is equal to original mean multiplied by the constant.
When every observation is multiplied by the same constant, the new sample standard deviation is equal to original standard deviation multiplied by the magnitude of the constant.

26. Normal (Gaussian) distribution
Mean and standard deviations are a particularly appropriate summary for data whose histogram approximates a normal distribution (the bell-shaped curve).
If you say that a set of data has a mean survival of 29 months, the typical listener will picture a bell-shaped curve centered with its peak at 29 months.

27. Rule of 3-sigma When data are approximately normally distributed:
approximately 68% of the data lie within one SD of the mean;
approximately 95% of the data lie within two SDs of the mean;
approximately 99% of the data lie within three SDs of the mean.

28. Normal (Gaussian) distribution
Central limit theorem:
Create a population with a known distribution that is not normal;
Randomly select many samples of equal size from that population;
Tabulate the means of these samples and graph the frequency distribution.
Central limit theorem states that if your samples are large enough, the distribution of the means will approximate a normal distribution even if the population is not Gaussian.
Mistakes:
Normal vs common (or disease free);
Few biological distributions are exactly normal.

29. Outliers
Values that lie very far away from the other values in the data set.

30. Outliers Outliers can occur for several reasons: Mistakes:
Invalid data entry
Biological diversity
Random chance
Experimental error
Skewed distribution
Mistakes:
Not realizing that outliers are common in data sampled from skewed distribution
Eliminating outliers only when you do not get the results you want

31. Outliers Outlier test:
If values are sampled from a normal distribution, what is the chance one value will be as far from the others as the extreme value observed?
Examples: Chauvenet criterion, Grubbs test, Peirce criterion
Nevertheless, deletion of outlier data is generally a controversial practice!

32. Inferential statistics
Population
Parameters
Sampling
From population to sample
Sample
Statistics
From sample to population
Inferential statistics

33. Confidence interval for the population mean
Population mean: point estimate vs interval estimate
Standard error of the mean – how close the sample mean is likely to be to the population mean.
Assumptions: a random representative sample, independent observations, the population is normally distributed (at least approximately).
Confidence interval depends on: sample mean, standard deviation, sample size, degree of confidence.
Mistakes:
95% of the values lie within the 95% CI;
A 95% CI covers the mean ± 2 SD.

34. Standard error of mean
The sample mean estimates individual values. The uncertainty with which this mean estimates individual values is given by the standard deviation.
The sample mean estimates the population mean. The uncertainty with which this mean estimates the population mean is given by the standard error of the mean.

35. Confidence interval for the population mean
The confidence interval for the mean gives us a range of values around the mean where we expect the “true” population mean is located.
95% confidence interval for the population mean is:

36. Confidence interval for the population mean
The duration of time from first exposure to HIV infection to AIDS diagnosis is called the incubation period. The incubation periods (in years) of a random sample of 30 HIV infected individuals are: 12.0, , 9.5, 6.3, 13.5, 12.5, 7.2, 12.0, 10.5, 5.2, 9.5, 6.3, 13.1, 13.5, , 10.7, 7.2, 14.9, 6.5, 8.1, 7.9, 12.0, 6.3, 7.8, 6.3, 12.5, 5.2, 13.1, , 7.2. Calculate the 95% CI for the population mean incubation period in HIV.
X = 9.5 years; SD = 2.8 years
SEM = 0.5 years
95% level of confidence => Z = 1.96
µ = 9.5 ± (1.96 x 0.5) = 9.5 ± 1 years
95% CI for µ is (8.5; 10.5 years)

37. Confidence interval for the population mean
X = 9.5 years; SD = 2.8 years
SEM = 0.5 years
95% level of confidence => Z = 1.96
µ = 9.5 ± (1.96 x 0.5) = 9.5 ± 1 years
95% CI for µ is (8.5; 10.5 years)
99% level of confidence => Z = 2.58
µ = 9.5 ± (2.58 x 0.5) = 9.5 ± 1.3 years
99% CI for µ is (8.2; 10.8 years)

38. Describing qualitative data
Improvement
No improvement
Total
Gluten-free diet 54 46
100
No gluten-free diet 47 53
101 99
200

39. Describing qualitative data
Standard error of proportion:
The 95% confidence interval for a population proportion is: