1 Stat 1510 Statistical Thinking & Concepts Describing Distributions with Numbers

2 Numerical Summaries u Center of the data –mean –median u Variation –range –quartiles (interquartile range) –variance –standard deviation

3 Mean or Average u Traditional measure of center u Sum the values and divide by the number of values

4 Median (M) u A resistant measure of the data’s center u At least half of the ordered values are less than or equal to the median value u At least half of the ordered values are greater than or equal to the median value u If n is odd, the median is the middle ordered value u If n is even, the median is the average of the two middle ordered values

5 Median (M) Location of the median: L(M) = (n+1)/2, where n = sample size. Example: If 25 data values are recorded, the Median would be the (25+1)/2 = 13 th ordered value.

6 Median u Example 1 data: Median (M) = 4 u Example 2 data: Median = 5 (ave. of 4 and 6) u Example 3 data: Median  2 (order the values: 2 4 6, so Median = 4)

7 Comparing the Mean & Median u The mean and median of data from a symmetric distribution should be close together. The actual (true) mean and median of a symmetric distribution are exactly the same. u In a skewed distribution, the mean is farther out in the long tail than is the median [the mean is ‘pulled’ in the direction of the possible outlier(s)].

8 Question A recent newspaper article in California said that the median price of single-family homes sold in the past year in the local area was $136,000 and the mean price was $149,160. Which do you think is more useful to someone considering the purchase of a home, the median or the mean?

9 Case Study Airline fares appeared in the New York Times on November 5, 1995 “...about 60% of airline passengers ‘pay less than the average fare’ for their specific flight.” u How can this be? 13% of passengers pay more than 1.5 times the average fare for their flight

10 Misuse of Statitics In 2003, President George W. Bush announced that under his policy, ”92 million Americans received an average tax cut of $1,083," those numbers were not, strictly speaking, incorrect. However, they camouflaged the fact that some 45 million people would receive less than $100 tax relief, whereas the top 1% of income earners were gifted a whopping $30,127.

11 Mode The mode is the value that occurs most frequently in a data set The mode of {2,3,4,4,4,5,5,7,10} is 4 For example, taking a sample of Korean family names, one might find that "Kim" occurs more often than any other name. Then "Kim" would be the mode of the sample. In any voting system, a single modal value determines the victor, while a multi- modal outcome would require some tie- breaking procedure to take place.

12 Spread, or Variability u If all values are the same, then they all equal the mean. There is no variability. u Variability exists when some values are different from (above or below) the mean. u We will discuss the following measures of spread: range, quartiles, variance, and standard deviation

13 Range u One way to measure spread is to give the smallest (minimum) and largest (maximum) values in the data set; Range = max  min u The range is strongly affected by outliers

14 Quartiles u Three numbers which divide the ordered data into four equal sized groups. u Q 1 has 25% of the data below it. u Q 2 has 50% of the data below it. (Median) u Q 3 has 75% of the data below it.

15 Quartiles Uniform Distribution Q1Q1 Q2Q2 Q3Q3

16 Obtaining the Quartiles u Order the data. u For Q 2, just find the median. u For Q 1, look at the lower half of the data values, those to the left of the median location; find the median of this lower half. u For Q 3, look at the upper half of the data values, those to the right of the median location; find the median of this upper half.

17 Weight Data: Sorted L(M)=(53+1)/2=27 L(Q 1 )=(26+1)/2=13.5

18 Weight Data: Quartiles u Q 1 = u Q 2 = 165 (Median) u Q 3 = 185

third quartile median or second quartile first quartile Weight Data: Quartiles

20 Five-Number Summary u minimum = 100 u Q 1 = u M = 165 u Q 3 = 185 u maximum = 260 Interquartile Range (IQR) = Q 3  Q 1 = 57.5 IQR gives spread of middle 50% of the data

21 Boxplot u Central box spans Q 1 and Q 3. u A line in the box marks the median M. u Lines extend from the box out to the minimum and maximum.

22 M Weight Data: Boxplot Q1Q1 Q3Q3 minmax Weight

23 Example from Text: Boxplots

24 Identifying Outliers u The central box of a boxplot spans Q 1 and Q 3 ; recall that this distance is the Interquartile Range (IQR). u We call an observation a suspected outlier if it falls more than 1.5  IQR above the third quartile or below the first quartile.

25 Variance and Standard Deviation u Recall that variability exists when some values are different from (above or below) the mean. u Each data value has an associated deviation from the mean:

26 Deviations u what is a typical deviation from the mean? (standard deviation) u small values of this typical deviation indicate small variability in the data u large values of this typical deviation indicate large variability in the data

27 Variance u Find the mean u Find the deviation of each value from the mean u Square the deviations u Sum the squared deviations u Divide the sum by n-1 (gives typical squared deviation from mean)

28 Variance Formula

29 Standard Deviation Formula typical deviation from the mean [ standard deviation = square root of the variance ]

30 Variance and Standard Deviation Example from Text Metabolic rates of 7 men (cal./24hr.) :

31 Variance and Standard Deviation Example from Text ObservationsDeviationsSquared deviations  1600 = 192 (192) 2 = 36,  1600 = 66 (66) 2 = 4,  1600 = -238 (-238) 2 = 56,  1600 = 14 (14) 2 =  1600 = -140 (-140) 2 = 19,  1600 = 267 (267) 2 = 71,  1600 = -161 (-161) 2 = 25,921 sum = 0sum = 214,870

32 Variance and Standard Deviation Example from Text

33 Choosing a Summary u Outliers affect the values of the mean and standard deviation. u The five-number summary should be used to describe center and spread for skewed distributions, or when outliers are present. u Use the mean and standard deviation for reasonably symmetric distributions that are free of outliers.

34 Number of Books Read for Pleasure: Sorted M L(M)=(52+1)/2=26.5

35 Five-Number Summary: Boxplot Median = 3 interquartile range (iqr) = = 4.5 range = 99-0 = 99 Mean = 7.06 s.d. = Number of books