1. How to calculate Confidence Intervals and Weighting Factors
Christina Blakey

2. Useful definitions
Weighting Factors make the achieved sample match the population
Grossing Factors make the sample the size of the population
Confidence Intervals show how accurate our estimates are

3. Weighting Factors For example, we have an unequal
number of male and female
respondents in our sample (e.g. 70%
female and 30% male) but we know
that the population is 50% female and
50% male.
In order to ensure that our results
represent the population each “male”
answer would be given more weight
than each “female” answer.

4. How do we calculate Weighting Factors?
Achieved Sample
Known Population
Weighting Factor
Men
150 (36%)
4500 (45%)
45/36 = 1.25
Women
270 (64%)
5400 (55%)
55/64 = 0.86
Total
420
9,900

5. Grossing Factors Otherwise known as population based weighting, grossing up weights
Form of weighting used to “gross up” results to the population being studied so we can make statements about the population rather than just the sample
The grossing factor is the known population divided
by the achieved sample size

6. Grossing Factors Achieved Sample Known Population Grossing Factor Men
Achieved Sample
Known Population
Grossing Factor
Men
150
4500
Women
270
5400
Total
420
9,900
9,900/420 = 23.6

7. Combining Weighting and Grossing Factors
Usually weighting and grossing factors are
used together
How do we combine them?
There are two ways
Both methods achieve the same result so
we will only focus on one

8. Weighting and Grossing Factors
Achieved Sample
Known Population
Weighting and Grossing Factor
Men
150
4500
4,500/150 = 30
Women
270
5400
5,400/270 = 20
Total
420
9,900

9. How to apply weights in practice?
How do we add this information to our data set?
There are a number of different statistical packages
available and each have different methods of adding
weights to the data.
The simplest way is too add a column to your data set
entitled weight.
Each individual response can then be multiplied by this
column.

10. Confidence Intervals (CI’s)
Confidence intervals are one of the most important ways that statisticians quantify the error in an estimate
They show how accurate our results are.
The narrower the intervals the more accurate our estimates are.

11. Calculating Confidence Intervals
So using our example from before
Our weighted result shows that 70% of people prefer dogs to cats.
How do we calculate a 95% Confidence Interval?
A 95% confidence interval is 1.96 multiplied by the
standard error
There are different ways to calculate Standard Error for
Means and Proportions

12. Calculating Confidence Intervals
The standard error for proportions is:
s.e = √ ((p (100-p)) / n)
(Where p is our result and n is our sample size)
So:
√ (( 70(100 – 70)) / 420)
√ ((70 * 31) /420)
√ (2100/420)
√ 5
s.e = 2.24

13. Calculating Confidence Intervals
CI = 2.24 * 1.96
CI = 4.4
70 – 4.4 = 65.6
= 74.4
Therefore we are 95% confident that the true percentage of people who prefer dogs to cats is between 65.6% and 74.4%

14. Variance, standard deviation and standard error
Variance = the sum of squared differences from the mean divided by n-1
Variance = 30 / 4 = 7.5
Standard error = the square root of the variance divided by the sample size
SE = √ (variance / n)
= √ (7.5 / 5)
= 1.22
Sample values (n=5)
Difference from mean
Squared difference from the mean
172
= -1
-1 x -1 = 1
169 2 4
168 3 9
175 -4 16
171
Mean = 171
Sum = 0
Sum of squares = 30

15. Confidence Intervals
So 1.96 times the standard error gives us the 95% confidence limits.
Our standard error is x 1.22 = 2.4
Our sample mean is – 2.4 = 168.6
= 173.4
We can be 95% confident that the true mean (the population mean) lies
between and

16. Confidence Interval Calculator

17. Any questions??