1 2.4 Describing Distributions Numerically – cont. Describing Symmetric Data

2 Symmetric Data Body temp. of 93 adults

3 Recall: 2 characteristics of a data set to measure n center measures where the “middle” of the data is located n variability measures how “spread out” the data is

4 Measure of Center When Data Approx. Symmetric n mean (arithmetic mean) n notation

5

6 Connection Between Mean and Histogram n A histogram balances when supported at the mean. Mean x = 140.6

7 Mean: balance point Median: 50% area each half right histo: mean yrs, median 57.7yrs

8 Properties of Mean, Median 1.The mean and median are unique; that is, a data set has only 1 mean and 1 median (the mean and median are not necessarily equal). 2.The mean uses the value of every number in the data set; the median does not.

9 Think about mean and median n Six people in a room have a median age of 45 years and mean age of 45 years. n One person who is 40 years old leaves the room. n Questions: 1.What is the median age of the 5 people remaining in the room? 2.What is the mean age of the 5 people remaining in the room? Can’t answer  6=270; =230; 230/5=46

10 Example: class pulse rates n

, 2014 baseball salaries n 2010 n = 845  = $3,297,828 median = $1,330,000 max = $33,000,000 n 2014 n = 848  = $3,932,912 median = $1,456,250 max = $28,000,000

12 Disadvantage of the mean n Can be greatly influenced by just a few observations that are much greater or much smaller than the rest of the data

13 Mean, Median, Maximum Baseball Salaries

14 Skewness: comparing the mean, and median n Skewed to the right (positively skewed) n mean>median

15 Skewed to the left; negatively skewed n Mean < median n mean=78; median=87;

16 Symmetric data n mean, median approx. equal

18 Describing Symmetric Data (cont.) n Measure of center for symmetric data: n Measure of variability for symmetric data?

19 Example n 2 data sets: x 1 =49, x 2 =51 x=50 y 1 =0, y 2 =100 y=50

20 On average, they’re both comfortable

21 Ways to measure variability 1. range=largest-smallest ok sometimes; in general, too crude; sensitive to one large or small obs.

22 Previous Example

23 The Sample Standard Deviation, a measure of spread around the mean n Square the deviation of each observation from the mean; find the square root of the “average” of these squared deviations

24 Calculations … Mean = 63.4 Sum of squared deviations from mean = 85.2 (n − 1) = 13; (n − 1) is called degrees freedom (df) s 2 = variance = 85.2/13 = 6.55 inches squared s = standard deviation = √6.55 = 2.56 inches Women height (inches)

25 1. First calculate the variance s Then take the square root to get the standard deviation s. Mean ± 1 s.d. We’ll never calculate these by hand, so make sure to know how to get the standard deviation using your calculator, Excel, or other software.

26 Population Standard Deviation

27 Remarks 1. The standard deviation of a set of measurements is an estimate of the likely size of the chance error in a single measurement

28 Remarks (cont.) 2. Note that s and  are always greater than or equal to zero. 3. The larger the value of s (or  ), the greater the spread of the data. When does s=0? When does  =0?

29 Remarks (cont.) 4. The standard deviation is the most commonly used measure of risk in finance and business –Stocks, Mutual Funds, etc. 5. Variance  s 2 sample variance   2 population variance  Units are squared units of the original data  square $, square gallons ??

30 Remarks 6):Why divide by n-1 instead of n? n degrees of freedom n each observation has 1 degree of freedom however, when estimate unknown population parameter like , you lose 1 degree of freedom

31 Remarks 6) (cont.):Why divide by n-1 instead of n? Example n Suppose we have 3 numbers whose average is 9 nx1=x2=nx1=x2= n then x 3 must be n once we selected x 1 and x 2, x 3 was determined since the average was 9 n 3 numbers but only 2 “degrees of freedom”

32 Computational Example

33 class pulse rates

34 Example #1#2#3# n x n s n m

35 Boxplots: same mean, standard deviation

36 More Boxplots of the 4 data sets

37 Review: Properties of s and  s and  are always greater than or equal to 0 when does s = 0?  = 0? The larger the value of s (or  ), the greater the spread of the data n the standard deviation of a set of measurements is an estimate of the likely size of the chance error in a single measurement

38 Summary of Notation