Populations, Samples, Standard errors, confidence intervals Dr. Omar Al Jadaan

Populations Population is a concept used to describe the entire group of individuals in whom we are interested. Parameters (mean, median standard deviation, rate, proportion) are quantities used to describe characteristics of such population. Sample is a subset of the population, which having representatives of that population.

Why we use sample We use the sample to – draw conclusions about the population. – Inference about the population as a whole. – Estimate the parameters of population like: Mean Standard deviation Proportion Rate

Sample Sample consist of all the members, then there is no bias in the population parameters. Random sampling  unbiased estimate Convenience sampling “ make-do”, apply the study on them when they are available, here we might get biased estimate

Standard error When the sample size is small then the uncertainty will be higher than when the sample size is large. Very uncertain when the sample size is small Less uncertain when the sample size is large The spread or variability of the sample mean reduces as the sample size increases.

The mean of all sample means will be the same as the population mean. The standard deviation of all the sample means known as the standard error (SE) of the mean or SEM Given a large enough sample size, the distribution of sample mean, will be roughly Normal regardless of the distribution of the variable

If you know the standard deviation of the population then the sample standard deviation is From the sample we can calculate the sample mean is the best estimation of µ (mean of the population). The same sample provides s, the standard deviation of the observations, as an estimate of σ.

With single study the investigator can estimate the standard deviation of the distribution of the mean by without having to repeat the study at all. The standard error provides a measure of precision of our sample estimate of the population.

Properties of standard error The standard error is a measure of the precision of a sample estimate. It provides a measure of how far from the true value in the population the sample estimate is likely to be. All standard errors have the following interpretation: – A large standard error indicate that the estimate is imprecise – A small standard error indicates that the estimate is precise. – The standard error is reduced, that is, we obtain a more precise estimate, if the size of the sample is increased.

Central Limit theorem The important point is whatever the parent distribution of a variable, the distribution of the sample means will be nearly Normal, as long as the samples are large enough

Standard error for proportion and rates DistributionParameterPopulation value Sample estimate Standard error Normalmeanµ BinomialProportionπP PoissonRateλr

Examples We have 124 patients have gone under acupuncture treatment for tension headache type with probability p=0.46 what is the standard error if you know that this sample follow the Binomial distribution.

Solution From the previous table we have SE(P)= then

Example Over a period of two years the rate of donation was r=1.82 per a day collected from 731 cases, assuming the Poisson distribution. What is the standard error.

What is the difference between STD and Standard error SE is always refer to an estimate of a parameter, where we get precise estimate when the number of observations get larger. Off course when SE value getting smaller. Standard deviation s is a measure of the variability between individuals with respect to the measurement under consideration. Whereas the standard error (SE) is a measure of uncertainty in the sample statistic.

Standard error of difference To compare two groups, we have to calculate the standard difference error between the groups DistributionParameterPopulation valueSample estimate Standard error Normalmeanµ 1 -µ 2 BinomialProportionπ1-π2π1-π2 P1-p2P1-p2 PoissonRateλ1-λ2λ1-λ2 r1-r2r1-r2

Example The results from the 148 patients are expressed using the group mean and standard deviation (SD) as follows: N int =93, m 1 =211, SD(int)=S int =118 N ctl =91, m 2 =91, SD(ctl)=S ctl =99 From these data d=m 1 -m 2 =211-91=88m The corresponding standard error is SE(d)

Confidence intervals for an estimate A confidence interval defines a range of values within which our population parameter likely to lie. It is termed as a95% confidence interval

Example From the birthweight of 98 infant, mean was 1.31kg and SD=0.42, so SE= kg The 95 CI(confidence interval) *0.04 to * to 1.93 The same is applicable to proportion and rate And the difference of means Calculate the confidence interval for the previous examples