Download
1 / 27
270 likes | 271 Views
Learn how to find the median, quartiles, and mean of a dataset, and understand their significance in describing distributions. Explore examples and key concepts in numerical summaries.
E N D
Chapter 12 Describing Distributions with Numbers Chapter 12
Median • Example 1 data: 2 4 6 Median (M) = 4 • Example 2 data: 2 4 6 8 Median = 5 (avg. of 4 and 6) • Example 3 data: 6 2 4 Median 2 (order the values: 2 4 6 , so Median = 4) Chapter 12
Example: Finding the median • Consider the following set of data: 16 19 24 25 25 33 33 34 34 37 37 40 42 45 45 46 46 49 73 • There are 19 observations, hence the median is the one in the middle, the 10th one. Hence the median of these data is 37. Chapter 12
Example: Finding the median • Now consider the following data where there is an even number of observations: 3 9 9 22 29 32 32 33 39 39 42 49 52 58 65 70 • The ones in the middle are 33 and 39, and hence the median is the average of these values, 36. Chapter 12
Quartiles • Three numbers that divide the ordered data into four equal-sized groups. • Q1 has 25% of the data below it. • Q2 has 50% of the data below it. (Median) • Q3 has 75% of the data below it. Chapter 12
Example: Finding the quartiles • Consider the following set of data: 16 19 24 25 25 33 33 34 34 37 37 40 42 45 45 46 46 49 73 • We already know that the median is 37. To find the first quartile we consider the first half of this data, namely, 16 19 24 25 25 33 33 34 34 • There are 9 observations here, so the median is the 5th one, namely, 25. This is the first quartile. • To find the third quartile we consider the second half of the original data, that is, 37 40 42 45 45 46 46 49 73 • There are again 9 observations and the median is the one in the middle which is 45, this is the third quartile. Chapter 12
Example: Finding the quartiles • Next, consider the following set of data: 3 9 9 22 29 32 32 33 39 39 42 49 52 58 65 70 • We already know that the median is 36. To find the first quartile we consider the first half of this data, namely, 3 9 9 22 29 32 32 33 • There are 8 observations here, so the median is the average of the 4th and 5th one, namely, 25.5. This is the first quartile. • To find the third quartile we consider the second half of the original data, that is, 39 39 42 49 52 58 65 70 • There are again 8 observations and the median is again the average of 4th and 5th observations, that is 50.5, this is the third quartile. Chapter 12
Example: Five-number summary and boxplot • Here are Roger Maris’ home run counts for his 12 years in the Major Leagues, arranged in order: 5 8 9 13 14 16 23 26 28 33 39 61 • Since there are an even number of observations, the median is the average of two middle values, that is, (16+23)/2 = 19.5 • The first quartile is the median of 5 8 9 13 14 16 that is (9+13)/2 = 11 • The third quartile is the median of the second half, 23 26 28 33 39 61 that is (28+33)/2 = 30.5 • The minimum and maximum values are obviously 5 and 61, respectively. • Hence a five-number summary of these data is 5 11 19.5 30.5 61 Chapter 12
Example: Five-number summary and boxplot • How does the boxplot of this distribution compare with those of the data corresponding to Barry Bonds and Mark McGwire given in the following figure: Chapter 12
Example: Education and Income • Interpret the following figure which contains the boxplots of income among adults with different levels of education. Chapter 12
Average or Mean Variance Chapter 12
Comparing the Mean & Median • 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. • 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)]. Chapter 12
Example: Computing the mean and standard deviation • Consider the following data 16 25 24 19 33 25 34 46 37 33 42 40 37 34 49 73 46 45 45 • The mean is Chapter 12
Example: Computing the mean and standard deviation • The previous figure shows Barry Bonds’s home run counts, with their mean and distance of one observation from the mean indicated. • The idea behind the standard deviation is to average these 19 distances. To find the standard deviation we can use the following table Chapter 12
Example: Computing the mean and standard deviation The average of these distances (the variance) is: Notice that we “average” by dividing by one less than the number of observations. Finally, the standard deviation is computed by taking the square root of the variation: Chapter 12
Key Concepts • Numerical Summaries • Center (mean, median) • Spread (variance, standard deviation, quartiles) • Five-number summary & Boxplots • Choosing mean versus median • Choosing standard deviation versus five-number summary Chapter 12
Exercise 12.7 • College tuitions. The following figure is a stemplot of the tuition charged by 121 colleges in Illinois. The stems are thousands of dollars and the leaves are hundreds of dollars. • Find the five-number summary of Illinois college tuitions. • Would the mean tuition be clearly smaller than the median, about the same, or larger? Chapter 12
Figure 11.7 Stemplot of the Illinois tuition and fee data. (Data from the Web site www.collegeillinois.com/en/collegefunding/costs.htm. This figure was created using the Minitab software package.)
Exercise 12.11 • The richest 1%. The distribution of individual incomes in the U.S. is strongly skewed to the right. In 2004, the mean and median incomes of the top 1% of Americans were $315,000 and $1,259,000. • Which of these numbers is the mean? Chapter 12
Exercise 12.31 • Raising Pay. A school system employs teachers at salaries between $30,000 and $60,000. The teachers’ union and the school board are negotiating the form of next year’s increase in the salary schedule. Suppose that every teacher is given a flat $1,000 raise. • How much will the mean salary increase? The median salary? • Will a flat $1,000 raise increase the spread as measured by the distance between the quartiles? • Will a flat $1,000 raise increase the spread as measured by the standard deviation of the salaries? Chapter 12
