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The Relation of the sample to the whole population
When you undertake a study, it is usually necessary to draw a sample from the study population. You will then describe the population on the basis of the information collected from the sample. In other words you will generalize the findings from the sample to the large population. Obviously it is necessary to select such a sample which is representative of the population from which it was drawn. Any value of variable obtained from the sample (e.g. sample mean), can be considered as an estimate of the population value.
For example if we took a sample of 120 women from Peshawar city and found that the mean height was 158 cm, another sample of 120 women from another corner was taken and we found that now the mean height calculated is 157 cm which is not identical to the previous mean which was 158 cm.
If you draw another sample and found it to be 156 cm, which again differs from both the previously taken sample, you may think that these samples are not the representative of the population of the women of Peshawar. This phenomenon is called sampling variation.
HOW TO DETERMINE THE EXTENT TO WHICH A SAMPLE REPRESENTS THE POPULATION AS A WHOLE:
To find out to what extent a particular sample value deviates from the population value, a range or an interval around the sample value can be worked out which will most probably contain the population value, this range or interval is called the confidence interval
Definition of Confidence Interval
A confidence interval is the interval or range of values which most likely encompasses the true population value. The lower and upper limit of this interval are termed confidence limits.
For example:
A 95 % confidence interval of 152 to 164 cm for mean height of a population of women means that you are 95 % certain that the real population mean, which you cannot know exactly unless you measure the heights of all women, lies between 152 to 164 cm (152 is the lower confidence limit and 164 is the upper confidence limit).
The calculation of a confidence interval takes into account the ‘Standard Error’. The standard error gives an estimate of the degree to which the sample mean varies from population mean.The standard error is computed on the basis of the standard deviation.
1. Now I will show how the standard error and 95 % confidence interval of a mean (for numerical data )
Standard Error and 95 % confidence interval for a mean.
When dealing with numerical data you may wish to estimate to what degree the sample mean varies from the population mean.
The standard error for mean is calculated by dividing the standard deviation by the square root of the sample size
Standard error = Standard Deviation /√n
It can be assumed , for a normally distribute variables, that approximately 95 % of all possible sample mean lie within two standard error of the population mean. In other words we can be 95 % sure that the population mean, of which we want to have the best possible estimate, lies within two standard error of our sample mean.
When describing variables statistically you usually present the calculated sample mean plus or minus two standard error. This is then called the 95 % Confidence Interval. It means that you are 95 % certain that the population mean is in this interval.
Suppose we had a random sample of 11 three years old children
The sample mean was 16 kg
Standard Deviation is 2 kg
The standard Error = Standard Deviation /√sample size
So Standard error = 2/√11 = 0.6 kg
The 95 % confidence interval is 16 plus minus (2 Standard Errors)
As Standard error is 0.6 and 2x 0.6= 1.2 kg
So 16 + 1.2 = 17.2 kg
And 16 minus 1.2 = 14.8 kg
The 95 % confidence interval is 14.8 to 17.2 kg
It means that we are 95 % certain that the mean weght of all the three years old children in your population lies between 14.8 to 17.2 kg
Note that larger the sample size the smaller the standard error and the narrower the confidence interval will be. Thus advantage of large sample is that the sample mean will be a better of the population mean.
In the above example if the sample size was 20 instead of 11 then
SEM = 2√20 = 0.45
And 95 % confidence interval for mean weght would have been 16+ (2x0.45) = 16.9
16_ (2x .45) = 15.1 kg
so we are 95 % sure that the population mean weght of three years old children lies between 15.1 to 16.9 kg
When you undertake a study, it is usually necessary to draw a sample from the study population. You will then describe the population on the basis of the information collected from the sample. In other words you will generalize the findings from the sample to the large population. Obviously it is necessary to select such a sample which is representative of the population from which it was drawn. Any value of variable obtained from the sample (e.g. sample mean), can be considered as an estimate of the population value.
For example if we took a sample of 120 women from Peshawar city and found that the mean height was 158 cm, another sample of 120 women from another corner was taken and we found that now the mean height calculated is 157 cm which is not identical to the previous mean which was 158 cm.
If you draw another sample and found it to be 156 cm, which again differs from both the previously taken sample, you may think that these samples are not the representative of the population of the women of Peshawar. This phenomenon is called sampling variation.
HOW TO DETERMINE THE EXTENT TO WHICH A SAMPLE REPRESENTS THE POPULATION AS A WHOLE:
To find out to what extent a particular sample value deviates from the population value, a range or an interval around the sample value can be worked out which will most probably contain the population value, this range or interval is called the confidence interval
Definition of Confidence Interval
A confidence interval is the interval or range of values which most likely encompasses the true population value. The lower and upper limit of this interval are termed confidence limits.
For example:
A 95 % confidence interval of 152 to 164 cm for mean height of a population of women means that you are 95 % certain that the real population mean, which you cannot know exactly unless you measure the heights of all women, lies between 152 to 164 cm (152 is the lower confidence limit and 164 is the upper confidence limit).
The calculation of a confidence interval takes into account the ‘Standard Error’. The standard error gives an estimate of the degree to which the sample mean varies from population mean.The standard error is computed on the basis of the standard deviation.
1. Now I will show how the standard error and 95 % confidence interval of a mean (for numerical data )
Standard Error and 95 % confidence interval for a mean.
When dealing with numerical data you may wish to estimate to what degree the sample mean varies from the population mean.
The standard error for mean is calculated by dividing the standard deviation by the square root of the sample size
Standard error = Standard Deviation /√n
It can be assumed , for a normally distribute variables, that approximately 95 % of all possible sample mean lie within two standard error of the population mean. In other words we can be 95 % sure that the population mean, of which we want to have the best possible estimate, lies within two standard error of our sample mean.
When describing variables statistically you usually present the calculated sample mean plus or minus two standard error. This is then called the 95 % Confidence Interval. It means that you are 95 % certain that the population mean is in this interval.
Suppose we had a random sample of 11 three years old children
The sample mean was 16 kg
Standard Deviation is 2 kg
The standard Error = Standard Deviation /√sample size
So Standard error = 2/√11 = 0.6 kg
The 95 % confidence interval is 16 plus minus (2 Standard Errors)
As Standard error is 0.6 and 2x 0.6= 1.2 kg
So 16 + 1.2 = 17.2 kg
And 16 minus 1.2 = 14.8 kg
The 95 % confidence interval is 14.8 to 17.2 kg
It means that we are 95 % certain that the mean weght of all the three years old children in your population lies between 14.8 to 17.2 kg
Note that larger the sample size the smaller the standard error and the narrower the confidence interval will be. Thus advantage of large sample is that the sample mean will be a better of the population mean.
In the above example if the sample size was 20 instead of 11 then
SEM = 2√20 = 0.45
And 95 % confidence interval for mean weght would have been 16+ (2x0.45) = 16.9
16_ (2x .45) = 15.1 kg
so we are 95 % sure that the population mean weght of three years old children lies between 15.1 to 16.9 kg
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