1. MAT 1000 Mathematics in Today's World

2. Last Time 1.Three keys to summarize a collection of data: shape, center, spread. 2.Can measure spread with the five- number summary. 3.The five-number summary can be represented visually by a boxplot, which is useful for making comparisons between distributions.

3. Today Another measurement for the spread of a distribution: the standard deviation. For a distribution of the correct shape, the two numbers mean and standard deviation give us more information than the whole five-number summary. These special shaped distributions are called normal distributions, and they are very common.

4. Standard deviation

7. Now we make a table Add up all the numbers in the last column.

8. Standard deviation Now we make a table

9. Standard deviation

14. How should we interpret the standard deviation? If the standard deviation is 0 then there is no deviation from the mean (all the data is equal) Otherwise, the standard deviation will be positive. The larger the value of the standard deviation, the more spread out the data.

15. Five-number summary and standard deviation We have two ways to measure the center and spread of a distribution: 1.The five-number summary 2.The mean and standard deviation. If the data is symmetric without many outliers, we will see that the mean and standard deviation give lots of information. If the data is not very symmetric, or has lots of outliers, the five-number summary is best.

16. Normal distributions The goal is to summarize large data sets. For a one number summary, measures of center like mean or median are the best we have, but no one number summary is very informative. It may be surprising, but for a large group of commonly occurring distributions, a two number summary can be quite informative. These distributions are called normal distributions.

17. Normal distributions Normal distributions all have a particular shape: fairly symmetric, one peak, few outliers, and a characteristic “bell” shape. The shape is easier to see with a smooth curve…

18. Normal distributions As a histogram.

19. Normal distributions As a smooth curve.

20. Normal distributions Both at once.

21. Normal distributions As a histogram.

22. Normal distributions As a smooth curve

23. Normal distributions Both at once.

24. Normal distributions