The document discusses sampling distributions, standard error, and confidence intervals, emphasizing how sampling error can quantify the variability of estimates from population parameters. It explains the central limit theorem, which states that sampling distributions are approximately normally distributed, and highlights the significance of confidence intervals in estimating population characteristics with specified precision. Additionally, it provides formulas for calculating standard error and confidence intervals, stressing their importance in health research for accurate generalization of results.

1. Sampling
Distributions, Standard
Error, Confidence Interval
Oyindamola Bidemi YUSUF
KAIMRC-WR

3. SAMPLE
 Why do we sample?
 Note: information in sample may not
fully reflect what is true in the
population
 We have introduced sampling error by
studying only some of the population
 Can we quantify this error?

4. SAMPLING VARIATIONS
 Taking repeated samples
 Unlikely that the estimates would be exactly
the same in each sample
 However, they should be close to the true
value
 By quantifying the variability of these
estimates, precision of estimate is obtained.
 Sampling error is thereby assessed.

5. SAMPLING DISTRIBUTIONS
 Distribution of sample estimates
- Means
- Proportions
- Variance
 Take repeated samples and calculate
estimates
 Distribution is approximately normal

6.  Mathematicians have examined the
distribution of these sample estimates
and their results are expressed in the
central limit theorem

7. central limit theorem
 Sampling distributions are approximately normally
distributed regardless of the nature of the variable in
the parent population
 The mean of the sampling distribution is equal to the
true population mean
 Mean of sample means is an unbiased estimate of
the true population mean
 The standard deviation (SD) of sampling distribution
is directly proportional to the population SD and
inversely proportional to the square root of the
sample size

8. SUMMARY: DISTRIBUTION OF
SAMPLE ESTIMATES
 NORMAL
 Mean = True population mean
 Standard deviation = Population standard
deviation divided by square root of sample
size
 Standard deviation called standard error

9. ESTIMATION
 A major purpose or objective of health
research is to estimate certain population
characteristics or phenomena
 Characteristic or phenomenon can be
quantitative such as average SYSTOLIC
BLOOD PRESSURE of adult men or qualitative
such as proportion with MALNUTRITION
 Can be POINT or INTERVAL ESTIMATE

10. Point estimates
 Value of a parameter in a population
e.g. mean or a proportion
 We estimate value of a parameter using
using data collected from a sample
 This estimate is called sample statistic
and is a POINT ESTIMATE of the
parameter i.e. it takes a single value

11. STANDARD ERROR
 Used to describe the variability of
sample means
 Depends on variability of individual
observations and the sample size
 Relationship described as –
Standard error = Standard Deviation
Square root of sample
size

12. Sample 1 Mean
Sample 2 Mean
Sample 3 Mean
……….
….........
Sample n Mean
Standard error
Mean of the means
Mean of the means
This mean will also have a standard deviation= SE
Standard error

13. Standard Deviation or Standard
Error?
 Quote standard deviation if interest is in the
variability of individuals as regards the level
of the factor being investigated – SBP, Age
and cholesterol level.
 Quote standard Error if emphasis is on the
estimate of a population parameter.
It is a measure of uncertainty in the sample
statistic as an estimate of population
parameter.

14. Interpreting SE
 Large SE indicates that estimate is
imprecise
 Small SE indicates that estimate is
precise
 How can SE be reduced?

15. Answer
 If sample size is increased
 If data is less variable

16. INTERVAL ESTIMATE
 Is SE particularly useful?
 More helpful to incorporate this measure of
precision into an interval estimate for the
population parameter
 How?
 By using the knowledge of the theoretical
probability distribution of the sample statistic to
calculate a CI

17.  Not sufficient to rely on a single
estimate
 Other samples could yield plausible
estimates
 Comfortable to find a range of values
within which to find all possible mean
values

18. WHAT IS A CONFIDENCE
INTERVAL?
 The CI is a range of values, above and below a
finding, in which the actual value is likely to fall.
 The confidence interval represents the accuracy or
precision of an estimate.
 Only by convention that the 95% confidence level
is commonly chosen.
 Researchers are confident that if other surveys
had been done, then 95 per cent of the time — or
19 times out of 20 — the findings would fall in this
range.

19. CONFIDENCE INTERVAL
 Statistic + 1.96 S.E. (Statistic)
 95% of the distribution of sample
means lies within 1.96 SD of the
population mean

20. Interpretation
 If experiment is repeated many times,
the interval would contain the true
population mean on 95% of occasions
 i.e. a range of values within which we
are 95% certain that the true
population mean lies

21. Issues in CI interpretation
 How wide is it? A wide CI indicates that
estimate is imprecise
 A narrow one indicates a precise
estimate
 Width is dependent on size of SE, which
in turn depends on SS

22. Factors affecting CI
 A narrow or small confidence interval
indicates that if we were to ask the same
question of a different sample, we are
reasonably sure we would get a similar result.
 A wide confidence interval indicates that we
are less sure and perhaps information needs
to be collected from a larger number of
people to increase our confidence.

23.  Confidence intervals are influenced by
the number of people that are being
surveyed.
 Typically, larger surveys will produce
estimates with smaller confidence
intervals compared to smaller surveys.

24. Why are CIs important
 Because confidence intervals represent
the range of values scores that are
likely if we were to repeat the survey.
 Important to consider when
generalizing results.
 Consider random sampling and
application of correct statistical test
 Like comfort zones that encompass the
true population parameter

25. Calculating confidence limits
 The mean diastolic blood pressure from
16 subjects is 90.0 mm Hg, and the
standard deviation is 14 mm Hg.
Calculate its standard error and 95%
confidence limits.

26. Standard error = Standard Deviation
Square root of sample
size
14
√16

27.  95% CI: Statistic + 1.96 S.E. (Statistic)

28. ANWERS
 Standard error – 3.5
 95% confidence limits – 82.55 to 97.46

29. CI for a proportion
 P + 1.96 S.E. (P)
 SE(P)= √p(1-p)/n
 Online calculators are available

30. In summary
 SD versus SE
 Meaning and interpretation of CI
 Shopping for the right sampling
distribution

31.  THANK YOU