Solutions: Introduction to Statistics
You can find complete Learning Objectives for this unit of study here.
Range
| Exercise | Problem | Solution |
| 1 | Find the range of these distances run by 6 marathon runners: 10 km, 15 km, 12 km, 14 km, 8 km, 16 km |
8 km, 10 km, 12 km, 14 km, 15 km, 16 km 16 km – 8 km = 8 km |
| 2 | Find the range of these quiz scores: 93, 79, 83, 89, 90, 71, 85 |
71, 79, 83, 85, 89, 90, 93 93 – 71 = 22 |
| 3 | Find the range of these race times given in seconds: 7.3 s, 8.4 s, 8.0 s, 7.5 s, 9.4 s, 8.7 s, 9.1 s |
7.3 s, 7.5 s, 8.0 s, 8.4 s, 8.7 s, 9.1 s, 9.4 s 9.4 s – 7.3 s = 2.1 s |
| 4 | Employees at a retail store are paid the hourly wages listed below. What is the range of these hourly wages? $7.50, $9.25 $8.75 $9.50, $7.25 $8.50 |
$7.25, $7.50, $8.50, $8.75, $9.25, $9.50 $9.50 – 7.25 = $2.25 |
| 5 | In a game, points were won and lost, represented by integers. What is the range of points given below? –14, +21, –17, +25, 0, –19, +11, –20, +18 |
–20, –19, –17, –14, 0, +11, +18 +21, +25 +25 – –20 = +45, or 45 |
Mean
| Exercise | Problem | Solution |
| 1 | Find the mean of the whole numbers listed below. 1, 8, 7, 6, 8, 3, 2, 5, 4, 5 |
1 + 8 + 7 + 6 + 8 + 3 + 2 + 5 + 4 + 5 = 49 49 ÷ 10 = 4.9 |
| 2 | Find the mean of the decimals listed below. 5.3, 5.5, 2.2, 4.8, 3.2 |
5.3 + 5.5 + 2.2 + 4.8 + 3.2 = 21 21 ÷ 5 = 4.2 |
| 3 | Find the mean rounded to the nearest tenth. 0.34, 0.12, 0.48, 0.56, 0.71, 0.8, 0.65, 0.82 |
0.34 + 0.12 + 0.48 + 0.56 + 0.71 + 0.8 + 0.65 + 0.82 = 4.48 4.48 ÷ 8 = 0.56 0.56 rounded to the nearest tenth is 0.6, or .6 |
| 4 | Employees at a retail store are paid the hourly wages listed below. What is the mean hourly wage? $7.50, $9.25, $8.75, $9.50, $7.25, $8.75 |
$7.50 + $9.25 + $8.75 + $9.50 + $7.25 + $8.75 = $51.00 $51.00 ÷ 6 = $8.50 |
| 5 | What is the mean test score? 83, 71, 91, 79, 87, 80, 95, 79, 91, 85, 94, 73 |
83 + 71 + 91 + 79 + 87 + 80 + 95 + 79 + 91 + 85 + 94 + 73 = 1,008 1,008 ÷ 12 = 84 |
Non-Routine Mean
| Exercise | Problem | Solution | ||||||
| 1 | The mean of 5 hourly wages is $5.95. What is the sum of these wages? | 5 x $5.95 = $29.75 | ||||||
| 2 | The mean of a set of numbers is 123. The sum of the numbers is 2,214. How many numbers are in the set? | 2,214 ÷ 123 = 18 There are 18 numbers in the set. |
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| 3 | Laura’s math test grades are 87, 93, 89, and 85. What grade must she get on the fifth test in order to get a mean of 90 for the term? |
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| 4 | A small company has a limited budget for salaries. It can afford to pay an average of $35,000 a year to its employees. If the first 5 employees are paid $37,000, $38,000, $33,000, $39,000 and $29,000, then how much money can they pay the sixth employee without exceeding their budget? |
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| 5 | In problem 4, what would the mean salary be if the sixth was paid $40,000? | $37,000 + $38,000 + $33,000 + $39,000 + $29,000 + $40.000 = $216,000 $216,000 ÷ 6 = $36,000 Answer: 36000 |
Median
| Exercise | Problem | Solution |
| 1 | Find the median weight of 5 seventh grade students whose weights in pounds are listed below. 82 lb, 102 lb, 75 lb, 91 lb, 89 lb |
75 lb, 82 lb, 89 lb, 91 lb, 102 lb Median = 89 lb |
| 2 | The ages of 10 college students are listed below. Find the median. 18, 24, 20, 35, 19, 23, 26, 23, 19, 20 |
18, 19, 19, 20, 20, 23, 23, 24, 26, 35 20 + 23 = 43 43 ÷ 2 = 21.5 Median = 21.5 |
| 3 | Seven teenagers receive the allowances listed below. What is the median allowance? $15, $10, $12, $18, $14, $15, $8 |
$8, $10, $12, $14, $15, $15, $18 Median = $14 |
| 4 | Find the median number of magazines purchased in a store by 6 customers. 1, 7, 4, 2, 3, 4 |
1, 2, 3, 4, 4, 7 Median = 3.5 |
| 5 | The annual snowfall in inches for New York City for the last 9 years is listed below. Find the median snowfall. 20.5 in, 26.1 in, 32.5 in, 18.9 in, 33.4 in, 29.7 in, 19.8 in, 25.6 in, 34.3 in |
18.9 in, 19.8 in, 20.5 in, 25.6 in, 26.1 in, 29.7 in, 32.5 in, 33.4 in, 34.3 in Median = 26.1 in |
Mode
| Exercise | Problem | Solution |
| 1 | The test scores of 9 seventh grade students are listed below. Find the mode. 82, 92, 75 , 91, 92, 89, 95, 100, 86 |
75, 82, 86, 89, 91, 92, 92, 95, 100 Mode = 92 |
| 2 | The typing speeds of 12 secretaries are recorded below (in words per minute). Find the mode. 35, 43, 39, 46, 43, 47, 38, 51, 43, 38, 40, 45 |
35, 38, 38, 39, 40, 43, 43, 43, 45, 46, 47, 51 Mode = 43 |
| 3 | The manager of a video shop recorded the number of blank tapes sold per day in 2 weeks (below). Which of the following statements is true? 132, 121, 119, 116, 130, 121, 131, 117, 119, 135, 121, 129, 119, 134There is no mode. The mode is 119. The mode is 131. The modes are 119 and 121. |
116, 117, 119, 119, 119, 121, 121, 121, 129, 130, 131, 132, 134, 135, The modes are 119 and 121. |
| 4 | Ten earthquakes were measured using the Richter scale, and their magnitudes are listed below. Which of the following statements is true? 7.0, 6.2, 7.7, 8.0, 6.4, 7.2, 5.4, 6.6, 7.5, 5.9There is no mode. The mode is 6.4. The mode is 5.4. The data set is bimodal. |
5.4, 5.9, 6.2, 6.4, 6.6, 7.0, 7.2, 7.5, 7.7, 8.0 There is no mode. |
| 5 | Eleven daily temperatures for Juneau, Alaska are listed below, in degrees Fahrenheit: –21, 0, +2, –15, 0, +5, –18, +7, 0, –32, +21There is no mode. The mode is 0. The mode is –21. The data set is bimodal. |
–32, –21, –18, –15, 0, 0, 0, +2, +5, +7, +21 The mode is 0. |
Practice Exercises
| Exercise | Problem | Solution |
| 1 | Twelve students were given a math test, and the times (in minutes) to complete it are listed below. Find the range of these times. 10, 9, 12, 11, 8, 15, 9, 7, 8, 6, 12, 10 |
6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 15 15 – 6 = 9 Range = 9 |
| 2 | A relay race was completed by 7 participants, and their race times are given below (in seconds). What is the range of race times? 13.2, 14.5, 12.9, 13.9, 15.6, 14.1, 12.3 |
12.3, 12.9, 13.2, 13.9, 14.1, 14.5, 15.6 15.6 – 12.3 = 3.3 Range = 3.3 |
| 3 | What is the mean test score for the 6 scores listed below? 89, 93, 87, 86, 85, 94 |
89 + 93 + 87 + 86 + 85 + 94 = 534 534 ÷ 6 = 89 Mean = 89 |
| 4 | The average annual wind speed for the 5 windiest cities in the U.S. is given below in miles per hour. What is the mean of these annual wind speeds? 15.4, 14.0, 13.5, 13.1, 12.9 |
15.4 + 14.0 + 13.5 + 13.1 + 12.9 = 68.9 68.9 ÷ 5 = 13.78 Mean = 13.78 mph |
| 5 | What is the median age of the 7 children whose ages are listed below? 11, 10, 13, 11, 12, 16, 14 |
10, 11, 11, 12, 13, 14, 16 Median = 12 |
| 6 | The number of tornadoes that have occurred in the U.S. in the last 8 years is listed below. Find the median. 684, 764, 1133, 656, 702, 1303, 856, 1132 |
656, 684, 702, 764, 856, 1133, 1132, 1303 (764 + 856) ÷ 2 = 810 Median = 810 |
| 7 | Given below are the daily high temperatures for one winter week in Albany, New York. What is the mode of these temperatures? 39, 42, 34, 37, 39, 44, 41 |
34, 37, 39, 39, 41, 42, 44 Mode = 39 |
| 8 | Find the mode of the following integers. –18, –13, 0, +16, +3, –15, 0 , +11, –14, 0, +12 |
–18, –15, –14, –13, 0, 0, 0, +3, +11, +12, +16 The mode is 0 |
| 9 | Heather’s test scores are 81, 93, 74 and 95. What score must she get on the fifth test in order to get a mean of 85 on all five tests? | (81 + 93 + 74 + 95 + x) ÷ 5 = 85 343 + x = (5)(85) 343 + x = 425 x = 82 Heather needs an 82 on her fifth test |
| 10 | The mean of a set of numbers is 108. The sum of the numbers is 2,052. How many numbers are in the set? | 2052 ÷ 108 = 19 There are 19 numbers in the set. |
Challenge Exercises
| Exercise | Problem | Solution |
| 1 | The weights of 9 students, measured in pounds, are recorded below. Find the mean weight. 135, 120, 116, 119, 121, 125, 135, 131, 123 |
135 + 120 + 116 + 119 + 121 + 125 + 135 + 131 + 123 = 1,125 1,125 ÷ 9 = 125 |
| 2 | What is the range of weights given in problem 1? | 116, 119, 120, 121, 123, 125, 131, 135, 135 135 – 116 = 19 Range = 19 |
| 3 | What is the mode of the weights given in problem 1? | 116, 119, 120, 121, 123, 125, 131, 135, 135 Mode = 135 |
| 4 | What is the median of the weights given in problem 1? | 116, 119, 120, 121, 123, 125, 131, 135, 135 Median = 123 |
| 5 | The mean price of 5 items is $7.00. The prices of the first four items are $6.50, $8.00, $5.50 and $6.00. How much does the fifth item cost? | $6.50 + $8.00 + $5.50 + 6.00 + x = (5) ($7.00) $26.00 + x = $35.00 x = $35.00 – $26.00 = $9.00 The fifth item costs $9.00. |
| 6 | The range of a set of numbers is 1,362. The greatest number is 2,172. What is the least number? | 2,172 – 1,362 = 810 The least number is 810. |
| 7 | The mean of a set of 7 numbers is 13. What is the sum of the numbers? | 7 · 13 = 91 The sum of the numbers is 91. |
| 8 | Find the mode of the following test scores: 89, 78, 91, 82, 75, 89, 84, 95, 89, 93 | 75, 78, 82, 84, 89, 89, 89, 91, 93, 95 The mode is 89. |
| 9 | The mean of a set of data is 174.25 and the sum of the data is 1,394. How many numbers are in the set? | 1,394 ÷ 174.25 = 8 There are 8 numbers in the set. |
| 10 | The grade point averages of 10 students are listed below. Find the median grade point average. 3.15, 3.62, 2.54, 2.81, 3.97, 1.85, 1.93, 2.63, 2.50, 2.80 | 1.85, 1.93, 2.50, 2.54, 2.63, 2.80, 2.81, 3.15, 3.62, 3.97 2.63 + 2.80 = 5.43 5.43 ÷ 2 = 2.715 The median is 2.715. |