Chapter 3.4

Which of the following are resistant measures of​ dispersion?

Interquartile range

The table shows the weekly income of 20 randomly selected​ full-time students. If the student did not​ work, a zero was entered. ​(a) Check the data set for outliers. ​(b) Draw a histogram of the data. ​(c) Provide an explanation for any outliers. 313 75 119 0 315 337 0 119 395 0 419 449 3526 0 98 537 423 133 0 473

(a) The​ outlier(s) is/are 3526 (b) Not symetrical, 0 is the mode, 3526 outlier (c) Possible reasons for the outlier: a. A student with unusually high income b. A student providing false information c. Data entry error

Interpreting Negative z-Scores

With negative z-scores, we need to be careful when deciding the better outcome. For example, suppose Bob and Mary ran a marathon. Bob finished the marathon in 213 minutes, where the mean finishing time among all men was 241 minutes with a standard deviation of 57 minutes. Mary finished the marathon in 239 minutes, where the mean finishing time among all women was 273 minutes with a standard deviation of 52 minutes. Bob's z-score= -0.49 Mary's z-score= -0.65 Therefore, Mary did relatively better. Even though Bob's z-score is larger, Mary did better because her time is more standard deviations below the mean

z-score

a measure of how many standard deviations you are away from the norm (average or mean)

Suppose the data represent the inches of rainfall in April for a certain city over the course of 20 years. Given the quartiles Q1=1.660​, Q2=3.370​, and Q3=4.570​, compute the interquartile​ range, IQR.

IQR= 2.91

z-score relative to mean

If a data value is larger than the mean, the z-score is positive. If a data value is smaller than the mean, the z-score is negative. If the data value equals the mean, the z-score is zero. A z-score measures the number of standard deviations an observation is above or below the mean. For example, a z-score of 1.24 means the data value is 1.24 standard deviations above the mean. A z-score of −2.31 means the data value is 2.31 standard deviations below the mean. The z-score is unitless. It has mean 0 and standard deviation 1.

The​ _______ represents the number of standard deviations an observation is from the mean.

z-score

Which of the following are resistant measures of central​ tendency?

Median

Steps to finding quartiles

Step 1. Arrange the data in ascending order. Step 2. Determine the median, M, or second quartile, Q2. Step 3. Divide the data set into two halves: the observations less than M and the observations greater than M. The first quartile, Q1, is the median of the bottom half, and the third quartile, Q3, is the median of the top half. Do not include M in these halves.

Checking for Outliers by Using Quartiles

Step 1. Determine the first and third quartiles of the data. Step 2. Compute the interquartile range. Step 3. Determine the fences. Fences serve as cutoff points for determining outliers. Lower fence= Q1−1.5(IQR) Upper fence= Q3+1.5(IQR) Step 4. If a data value is less than the lower fence or greater than the upper fence, it is considered an outlier.

The following data represent the monthly phone​ use, in​ minutes, of a customer enrolled in a fraud prevention program for the past 20 months. The phone company decides to use the upper fence as the cutoff point for the number of minutes at which the customer should be contacted. What is the cutoff​ point? 344 543 420 303 525 370 466 431 509 375 341 536 540 471 373 453 310 549 343 398

The cutoff point is 757 minutes

Explain the circumstances for which the interquartile range is the preferred measure of dispersion. What is an advantage that the standard deviation has over the interquartile​ range?

The interquartile range is preferred when the data are skewed or have outliers. An advantage of the standard deviation is that it uses all the observations in its computation.

Inerquartile range IQR

The interquartile range, IQR, is the range of the middle 50% of the observations in a data set. That is, the IQR is the difference between the first and third quartiles and is found using this formula IQR=Q3−Q1.

In a certain​ city, the average​ 20- to​ 29-year old man is 69.8 inches​ tall, with a standard deviation of 3.0 inches, while the average​ 20- to​ 29-year old woman is 64.1 inches​ tall, with a standard deviation of 3.9 inches. Who is relatively​ taller, a​ 75-inch man or a​ 70-inch woman? Find the corresponding​ z-scores. Who is relatively​ taller, a​ 75-inch man or a​ 70-inch woman? Select the correct choice below and fill in the answer boxes to complete your choice. ​(Round to two decimal places as​ needed.) A. The​ z-score for the woman​, nothing​, is larger than the​ z-score for the man​, nothing​, so she is relatively taller. B. The​ z-score for the man​, nothing​, is smaller than the​ z-score for the woman​, nothing​, so he is relatively taller. C. The​ z-score for the woman​, nothing​, is smaller than the​ z-score for the man​, nothing​, so she is relatively taller. D. The​ z-score for the man​, 1.731.73​, is larger than the​ z-score for the woman​, 1.511.51​, so he is relatively taller.

The​ z-score for the man​, 1.73​, is larger than the​ z-score for the woman​, 1.51, so he is relatively taller.

A manufacturer of bolts has a​ quality-control policy that requires it to destroy any bolts that are more than 4 standard deviations from the mean. The​ quality-control engineer knows that the bolts coming off the assembly line have mean length of 8 cm with a standard deviation of 0.05 cm. For what lengths will a bolt be​ destroyed?

A bolt will be destroyed if the length is less than 7.8 cm or greater than 8.2 cm. = xbar +/- SDev. from mean (SDev given) =(xbar+SD from mean (SD), xbar-SD from mean (SD) =(... , ...) less than or greater than

Outliers

Outliers can occur by chance, because of errors in the measurement of a variable, during data entry, or from errors in sampling. Sometimes extreme observations are common within a population.

The accompanying data represent the monthly rate of return of a certain​ company's common stock for the past few years. ​(a) Determine and interpret the quartiles. ​(b) Check the data set for outliers. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. 0.26 0.26 0.03 0.06 0.06 −0.04 −0.04 0.22 0.46 0.06 −0.15 0.19 0.05 0.18 0.09 0.02 −0.05 −0.02 0.08 0.02 −0.02 0.13 −0.08 -0.02 0.06 −0.01 0.07 −0.05 0.01 −0.10 0.02 0.03 0.01 0.11 −0.11 0.09 0.10 0.25 −0.02 0.03

Q1= -0.02 Q2= 0.03 Q3= 0.095 IQR= 0.115 Of the monthly​ returns, 25% are less than or equal to the first​ quartile, 50% are less than or equal to the second​ quartile, and​ 75% are less than or equal to the third quartile. The​ outlier(s) is/are 0.46

WHICH MEASURES TO REPORT

Shape of Distribution: a. Symmetric b. Skewed Left Or Skewed Right Measure of Central Tendency a. Mean b. Median Measure of Dispersion a. Standard Deviation b. Interquartile Range

The accompanying data represent the miles per gallon of a random sample of cars with a​ three-cylinder, 1.0 liter engine. ​(a) Compute the​ z-score corresponding to the individual who obtained 49.0 miles per gallon. Interpret this result. ​(b) Determine the quartiles. ​(c) Compute and interpret the interquartile​ range, IQR. ​(d) Determine the lower and upper fences. Are there any​ outliers? 32.3 34.4 34.6 35.3 36 36.4 37.5 37.6 37.8 38 38.1 38.5 38.6 38.8 39.4 39.5 40.3 40.6 41.4 41.7 42.6 42.7 43.3 49

(a) The​ z-score corresponding to the individual is 2.86 and indicates that the data value is 2.86 standard​ deviation(s) above the mean. (b) Q1= 36.95 Q2= 38.55 Q3= 41 (c) The interquartile range is 4.05 mpg. It is the range of the middle​ 50% of the observations in the data set. (d) Lower Fence= 30.875 Upper Fence= 47.075 (e) Outlier= 49

Suppose the data represent the inches of rainfall in April for a certain city over the course of 20 years. Given the quartiles Q1=2.180​, Q2=3.370​, and Q3=4.685​, determine the lower and upper fences. Are there any​ outliers, according to this​ criterion? 0.98 1.14 1.44 1.76 2.03 2.33 2.56 2.73 2.87 3.32 3.42 3.81 4.18 4.39 4.61 4.76 5.24 5.56 5.69 6.03

The lower fence is negative 1.578 The upper fence is 8.443 ​Noo outliers, all the values are between the lower and upper fences.

Things to remember

The median is resistant to extreme values, so it is the preferred measure of central tendency when data are skewed right or skewed left. The three measures of dispersion that are not resistant are the range, standard deviation, and variance. The interquartile range is resistant. However, the median, Q1, and Q3 do not provide information about the extremes of the data. For this, we need the smallest and largest values in the data set.

Quartiles

The most common percentiles are quartiles, which divide data sets into fourths, or four equal parts. - The first quartile, denoted Q1, divides the bottom 25% of the data from the top 75%. - The second quartile, Q2, divides the bottom 50% of the data from the top 50%. = the median -The third quartile, Q3, divides the bottom 75% of the data from the top 25%.

Violent crimes include​ rape, robbery,​ assault, and homicide. The following is a summary of the​ violent-crime rate​ (violent crimes per​ 100,000 population) for all states of a country in a certain year. Q1=271.8​, Q2=387.4​, Q3=528.3 Provide an interpretation of these results. Choose the correct answer below. A. 75% of the states have a​ violent-crime rate that is 271.8 crimes per​ 100,000 population or less.​ 50% of the states have a​ violent-crime rate that is 387.4 crimes per​ 100,000 population or less.​ 25% of the states have a​ violent-crime rate that is 528.3 crimes per​ 100,000 population or less. B. 25% of the states have a​ violent-crime rate that is 271.8 crimes per​ 100,000 population or more.​ 50% of the states have a​ violent-crime rate that is 387.4 crimes per​ 100,000 population or more.​ 75% of the states have a​ violent-crime rate that is 528.3 crimes per​ 100,000 population or more. C. 25% of the states have a​ violent-crime rate that is 271.8 crimes per​ 100,000 population or less.​ 50% of the states have a​ violent-crime rate that is 387.4 crimes per​ 100,000 population or less.​ 75% of the states have a​ violent-crime rate that is 528.3 crimes per​ 100,000 population or less.

C

quartile resisitance

Quartiles, however, are resistant. For this reason, quartiles are used to define a resistant measure of dispersion.

The mean finish time for a yearly amateur auto race was 187.55 minutes with a standard deviation of 0.397 minute. The winning​ car, driven by Sam​, finished in 186.82 minutes. The previous​ year's race had a mean finishing time of 111.7 with a standard deviation of 0.129 minute. The winning car that​ year, driven by Rita​, finished in 111.51 minutes. Find their respective​ z-scores. Who had the more convincing​ victory? Sam had a z-score of ______? Rita had a z-score of ______? Which driver had a more convincing​ victory? A. Sam had a more convincing victory because of a higher​ z-score. B. Sam had a more convincing victory because of a lower​ z-score. C. Rita a more convincing victory because of a lower​ z-score. D. Rita a more convincing victory because of a higher​ z-score.

Sam z-score= -1.84 Rita z-score= -1.47 Sam had a more convincing victory because of a lower​ z-score.

Reports

The direction describe the distribution will mean to describe its shape (skewed left, skewed right, or symmetric), its center (mean or median), and its spread (standard deviation or interquartile range).