Chapter 2 Describing Distributions with Numbers

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

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

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

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.

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)

Measure of center: the median The median is the midpoint of a distribution—the number such that half of the observations are smaller and half are larger. 1. Sort observations from smallest to largest. n = number of observations ______________________________ n = 24  n/2 = 12 Median = ( ) /2 = If n is even, the median is the mean of the two center observations  n = 25 (n+1)/2 = 26/2 = 13 Median = If n is odd, the median is observation (n+1)/2 down the list

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)].

Disease X: Mean and median are the same. Mean and median of a symmetric distribution Multiple myeloma: and a right-skewed distribution The mean is pulled toward the skew. Impact of skewed data

The median, on the other hand, is only slightly pulled to the right by the outliers (from 3.4 to 3.6). The mean is pulled to the right a lot by the outliers (from 3.4 to 4.2). Percent of people dying Mean and median of a distribution with outliers Without the outliers With the outliers

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?

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

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

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

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.

Quartiles Uniform Distribution Q1Q1 Q2Q2 Q3Q3

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.

M = median = 3.4 Q 1 = first quartile = 2.2 Q 3 = third quartile = 4.35 Measure of spread : quartiles The first quartile, Q 1, is the value in the sample that has 25% of the data at or below it. The third quartile, Q 3, is the value in the sample that has 75% of the data at or below it.

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.

M = median = 3.4 Q 3 = third quartile = 4.35 Q 1 = first quartile = 2.2 Largest = max = 6.1 Smallest = min = 0.6 “Five-number summary” Center and spread in boxplots

Example from Text: Boxplots

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:

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

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)

Variance Formula

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

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

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

Variance and Standard Deviation Example from Text

Notes on Standard Deviation u s measures spread about the mean, and should be used only if the mean is our measure of center. u In most cases s>0. s=0 where ________________. Cannot be less than 0. u As the spread of the data increases, s increases. u s has the same units as the original observations. u s is sensitive to skewness and outliers like the mean. u Sum of all deviations must be 0.

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.