How to calculate Confidence Intervals and Weighting Factors Christina Blakey 1 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 2 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. 3 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 4 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 5 Grossing Factors Achieved Sample Known Population Men 150 4500 Women 270 5400 Total 420 9,900 Grossing Factor 9,900/420 = 23.6 6 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 7 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 8 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. 9 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. 10 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 11 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 12 Calculating Confidence Intervals CI = 2.24 * 1.96 CI = 4.4 70 – 4.4 = 65.6 70 + 4.4 = 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% 13 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 172 171 - 172 = -1 -1 x -1 = 1 169 2 4 168 3 9 175 -4 16 171 0 0 Mean = 171 Sum = 0 Sum of squares =1430 Confidence Intervals So 1.96 times the standard error gives us the 95% confidence limits. Our standard error is 1.22. 1.96 x 1.22 = 2.4 Our sample mean is 171.0 171.0 – 2.4 = 168.6 171.0 + 2.4 = 173.4 We can be 95% confident that the true mean (the population mean) lies between 168.6 and 173.4. 15 Confidence Interval Calculator 16 Any questions?? • [email protected] • 0300 24 46792 17