1. Produced by MEI on behalf of OCR © OCR 2013 Introduction to Quantitative Methods Statistical Problem Solving Normal distribution summary Notes for students

2. Produced by MEI on behalf of OCR © OCR 2013 Contents Skew and symmetrical distributions Standard deviation Properties of Normal distributions Which Normal distribution? Fitting a Normal distribution Is it a good fit?

3. Produced by MEI on behalf of OCR © OCR 2013 Skewness A positively skewed distribution, with the tail to the right. A negatively skewed distribution. In this case, mean < median.

4. Produced by MEI on behalf of OCR © OCR 2013 Symmetrical distributions An approximately symmetrical distribution. The median is close to the mean. An approximately symmetrical distribution which can be modelled by a Normal distribution.

5. Produced by MEI on behalf of OCR © OCR 2013 Standard deviation Mean = 44.26, standard deviation = 19.75 Mean = 44.26, standard deviation = 11.39 The standard deviation is a measure of spread.

6. Produced by MEI on behalf of OCR © OCR 2013 Properties of Normal distributions A Normal distribution is a symmetrical, bell- shaped curve. 68% (about two-thirds) of the data lie within 1 standard deviation of the mean … continued

7. Produced by MEI on behalf of OCR © OCR 2013 …continued about 95% of the data lie within 2 sd of the mean nearly all the data lie within 3 sd of the mean

8. Produced by MEI on behalf of OCR © OCR 2013 Which Normal distribution? There are many different Normal distributions. If your dataset has mean 12 and standard deviation 3 then you would use the black Normal distribution to try to model it; this has mean (  ) 12 and standard deviation (  ) 3.

9. Produced by MEI on behalf of OCR © OCR 2013 Fitting a Normal distribution When you try to fit a Normal distribution, what you are really doing is saying something like: ‘I have a sample of 1174 birthweights. They come from a population of birthweights of all babies born in the USA. I will model the distribution of the population by a Normal distribution.’ The distribution of any sample from the population will be approximately symmetrical; approximately 95% of the data will be within 2 sd of the mean etc. For the Normal distribution which models the population, these will be exactly true.

10. Produced by MEI on behalf of OCR © OCR 2013 Is it a good fit? You can tell whether a Normal distribution is a good fit by looking at the curve drawn over the histogram, or by looking at a Normal probability plot. The closer the dots are to the straight line, the better the fit.

11. Produced by MEI on behalf of OCR © OCR 2013 Is it a good fit? Not everything is a Normal distribution!

12. Produced by MEI on behalf of OCR © OCR 2013 Acknowledgements Birthweight data from Child Health and Development Study School of Public Health, University of California, Berkeley http://www.stat.berkeley.edu/