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+ Chapter 1: Exploring Data Section 1.3 Describing Quantitative Data with Numbers

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+ Describing Quantitative Data Measuring Center: The Mean The most common measure of center is the ordinaryarithmetic average, or mean. Definition: To find the mean (pronounced “x-bar”) of a set of observations, add their values and divide by the number of observations. If the n observations are x 1, x 2, x 3, …, x n, their mean is: In mathematics, the capital Greek letter Σis short for “add them all up.” Therefore, the formula for the mean can be written in more compact notation:

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+ Example – Calculating the Mean How long do people spend traveling to work? Here are travel times in minutes for 15 randomly chosen workers in North Carolina. 1) Find the mean travel time for all 15 workers. 2) Calculate the mean again, this time excluding the person who reported a 60-minute travel time to work. What do you notice?

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+ Describing Quantitative Data Measuring Center: The Median Another common measure of center is the median. In section 1.2, we learned that the median describes themidpoint of a distribution. Definition: The median M is the midpoint of a distribution, the number such that half of the observations are smaller and the other half are larger. To find the median of a distribution: 1)Arrange all observations from smallest to largest. 2)If the number of observations n is odd, the median M is the center observation in the ordered list. 3)If the number of observations n is even, the median M is the average of the two center observations in the ordered list.

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+ Describing Quantitative Data Measuring Center People say that it takes a long time to work in New YorkState due to the heavy traffic near big cities. What do thedata say? Here are travel times in minutes of 20 randomlychosen New York workers. 103052540201015302015208515651560 4045 0 5 1 005555 2 0005 3 00 4 005 5 6 005 7 8 5 Key: 4|5 represents a New York worker who reported a 45- minute travel time to work.

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+ Comparing the Mean and the Median The mean and median measure center in different ways, andboth are useful. Don’t confuse the “average” value of a variable (the mean) with its“typical” value, which we might describe by the median. The mean and median of a roughly symmetric distribution are close together. If the distribution is exactly symmetric, the mean and median are exactly the same. In a skewed distribution, the mean is usually farther out in the long tail than is the median. Comparing the Mean and the Median Describing Quantitative Data

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+ Checkpoint Questions 1 through 4 refer to the following setting: Here is the stem plot of travel times to work for 20 randomly selected New Yorkers. Earlier, we found that the median was 22.5 minutes. 1) Based only on the stemplot, would you expect the mean travel time to be less than, about the same as, or larger than the median? Why? Since the distribution is skewed to the right, we would expect the mean to be larger than the median.

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+ 2) Use your calculator to find the mean travel time. Was your answer to Question 1 correct? 3) Interpret your result from Question 2 in context without using the words “mean” or “average.” If we divided the travel time up evenly among all 20 people, each would have a 31.25 minute travel time.

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+ 4) Would the mean or the median be a more appropriate summary of the center of this distribution of drive times? Justify your answer. In this case, since the distribution is skewed, the median is a better measure of the center of the distribution.

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+ Describing Quantitative Data Measuring Spread: The Interquartile Range ( IQR ) A measure of center alone can be misleading. A useful numerical description of a distribution requires both ameasure of center and a measure of spread. To calculate the quartiles: 1)Arrange the observations in increasing order and locate the median M. 2)The first quartile Q 1 is the median of the observations located to the left of the median in the ordered list. 3)The third quartile Q 3 is the median of the observations located to the right of the median in the ordered list. The interquartile range (IQR) is defined as: IQR = Q 3 – Q 1 How to Calculate the Quartiles and the Interquartile Range

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+ 510 15 20 2530 40 4560 6585 Describing Quantitative Data Find and Interpret the IQR 103052540201015302015208515651560 4045 Travel times to work for 20 randomly selected New Yorkers 510 15 20 2530 40 4560 6585 M = 22.5 Q 3 = 42.5 Q 1 = 15 IQR= Q 3 – Q 1 = 42.5 – 15 = 27.5 minutes Interpretation: The range of the middle half of travel times for the New Yorkers in the sample is 27.5 minutes.

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+ Describing Quantitative Data Identifying Outliers In addition to serving as a measure of spread, theinterquartile range (IQR) is used as part of a ruleof thumb for identifying outliers. Definition: The 1.5 x IQR Rule for Outliers Call an observation an outlier if it falls more than 1.5 x IQR above the third quartile or below the first quartile.

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+ Describing Quantitative Data In the New York travel time data, we found Q 1 =15 minutes, Q 3 =42.5 minutes, and IQR=27.5 minutes. For these data, 1.5 x IQR = 1.5(27.5) = 41.25 Q 1 - 1.5 x IQR = 15 – 41.25 = -26.25 Q 3 + 1.5 x IQR = 42.5 + 41.25 = 83.75 Any travel time shorter than -26.25 minutes or longer than 83.75 minutes is considered an outlier. 0 5 1 005555 2 0005 3 00 4 005 5 6 005 7 8 5 Example – Identifying Outliers Earlier, we noted the influence of one long travel time of 60 minutes in our sample of 15 North Carolina workers. Determine whether this value is an outlier.

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+ The Five-Number Summary The minimum and maximum values alone tell us little aboutthe distribution as a whole. Likewise, the median and quartilestell us little about the tails of a distribution. To get a quick summary of both center and spread, combineall five numbers. Describing Quantitative Data Definition: The five-number summary of a distribution consists of the smallest observation, the first quartile, the median, the third quartile, and the largest observation, written in order from smallest to largest. Minimum Q 1 M Q 3 Maximum

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+ Boxplots (Box-and-Whisker Plots) The five-number summary divides the distribution roughly intoquarters. This leads to a new way to display quantitative data,the boxplot. Draw and label a number line that includes the range of the distribution. Draw a central box from Q 1 to Q 3. Note the median M inside the box. Extend lines (whiskers) from the box out to the minimum and maximum values that are not outliers. How to Make a Boxplot Describing Quantitative Data

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+ Construct a Boxplot Consider our NY travel times data. Construct a boxplot. Example M = 22.5 Q 3 = 42.5Q 1 = 15Min=5 103052540201015302015208515651560 4045 510 15 20 2530 40 4560 6585 Max=85 Recall, this is an outlier by the 1.5 x IQR rule

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+ Checkpoint The 2009 roster of a professional football team included 10 offensive linemen. Their weights (in pounds) were: 1) Find the five-number summary for these data by hand. Show your work. 338318353313326 318307317311 Min = 307 Q1 = 311 Median = 317.5 Q3 = 326 Max = 353

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+ 2) Calculate the IQR. Interpret this value in context. 3) Determine whether there are any outliers using the 1.5 X IQR rule. The IQR is 326 – 311 = 15. This is the range of the middle half of the weights of offensive linemen in 2009. 1.5 X IQR = 1.5 (15) = 22.5. Any outliers occur below 288.5 (311 – 22.5) or above 348.5 (326 + 22.5). There are no observations below 288.5, but there is one observation above 348.5 – the value 353 is an outlier.

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+ 4) Draw a boxplot of the data. 310 320 330 340 350

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+ Describing Quantitative Data Measuring Spread: The Standard Deviation The most common measure of spread looks at how far eachobservation is from the mean. This measure is called thestandard deviation. Let’s explore it! Consider the following data on the number of pets owned bya group of 9 children. 1)Calculate the mean. 2)Calculate each deviation. deviation = observation – mean = 5 deviation: 1 - 5 = -4 deviation: 8 - 5 = 3

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+ Describing Quantitative Data Measuring Spread: The Standard Deviation xixi (x i -mean)(x i -mean) 2 11 - 5 = -4(-4) 2 = 16 33 - 5 = -2(-2) 2 = 4 44 - 5 = -1(-1) 2 = 1 44 - 5 = -1(-1) 2 = 1 44 - 5 = -1(-1) 2 = 1 55 - 5 = 0(0) 2 = 0 77 - 5 = 2(2) 2 = 4 88 - 5 = 3(3) 2 = 9 99 - 5 = 4(4) 2 = 16 Sum=? 3) Square each deviation. 4) Find the “average” squared deviation. Calculate the sum of the squared deviations divided by (n-1)…this is called the variance. 5) Calculate the square root of the variance…this is the standard deviation. “average” squared deviation = 52/(9-1) = 6.5 This is the variance. Standard deviation = square root of variance =

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+ Describing Quantitative Data Measuring Spread: The Standard Deviation Definition: The standard deviation s x measures the average distance of the observations from their mean. It is calculated by finding an average of the squared distances and then taking the square root. This average squared distance is called the variance.

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+ Checkpoint The heights (in inches) of five starters on a basketball team are 67, 72, 76, 76, and 84. 1) Find and interpret the mean. 2) Make a table that shows, for each value, its deviation from the mean and its squared deviation from the mean. The mean is 75. If the total of all the heights was the same, but all players were the same height, they would each be 75 inches tall. ObservationDeviationSquared Deviation 6767-75 = -8(-8) 2 = 64 7272-75 = -3(-3) 2 = 9 7676-75 = 1(1) 2 = 1 7676-75 = 1(1) 2 = 1 8484-75 = 9(9) 2 = 81 Total0156

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+ Checkpoint 3) Show how to calculate the variance and standard deviation from the values in your table. 4) Interpret the meaning of the standard deviation in this setting. The variance is the sum of the squared deviations (156) divided by 4 (the number of observations – 1). In this case, that means that the variance is 156/4 = 39 inches squared. The standard deviation is the square root of the variance, which is 6.24 inches. The players’ heights vary about 6.24 inches from the mean height of 75 inches on average.

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+ Choosing Measures of Center and Spread We now have a choice between two descriptions for centerand spread Mean and Standard Deviation Median and Interquartile Range The median and IQR are usually better than the mean and standard deviation for describing a skewed distribution or a distribution with outliers. Use mean and standard deviation only for reasonably symmetric distributions that don’t have outliers. NOTE: Numerical summaries do not fully describe the shape of a distribution. ALWAYS PLOT YOUR DATA! Choosing Measures of Center and Spread Describing Quantitative Data

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Due to the strong skewness and outliers, we’ll use the median and IQR, instead of the mean and standard deviation when discussing center and spread. Shape: Both distributions are heavily skewed right. Center: On average, females text more than males. The median number of texts for females (107) is about four times as high as for males (28). The median for the females is about the 3 rd quartile for the males. This indicates that over 75% of the males texted less than the “typical” (median) female.

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Spread: There is much more variability in texting among the females than the males (77). Outliers: There are two outliers in the male distribution: students who reported 213 and 214 texts in two days. The female distribution has no outliers. Conclusion: The data from this survey project give very strong evidence to support the students’ belief that females text more than males. Females sent and received a median of 107 texts over the two-day period, which exceeded the number of texts reported by over 75% of the males.

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+ In this section, we learned that… A numerical summary of a distribution should report at least its center and spread. The mean and median describe the center of a distribution in different ways. The mean is the average and the median is the midpoint of the values. When you use the median to indicate the center of a distribution, describe its spread using the quartiles. The interquartile range (IQR) is the range of the middle 50% of the observations: IQR = Q 3 – Q 1. Summary Section 1.3 Describing Quantitative Data with Numbers

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+ In this section, we learned that… An extreme observation is an outlier if it is smaller than Q 1 –(1.5xIQR) or larger than Q 3 +(1.5xIQR). The five-number summary (min, Q 1, M, Q 3, max) provides a quick overall description of distribution and can be pictured using a boxplot. The variance and its square root, the standard deviation are common measures of spread about the mean as center. The mean and standard deviation are good descriptions for symmetric distributions without outliers. The median and IQR are a better description for skewed distributions. Summary Section 1.3 Describing Quantitative Data with Numbers

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