Mathbench

Descriptive statistics: centre, spread and quantiles

The mean, median and mode, the range, variance, standard deviation and interquartile range, percentiles and quartiles, outliers and their effect on the mean, and the shape of a distribution. Set for OMPT-E.

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Questions

Sheet length
1

For the data 23,71,79,33,22,53,35,41,23, \allowbreak 71, \allowbreak 79, \allowbreak 33, \allowbreak 22, \allowbreak 53, \allowbreak 35, \allowbreak 41\mathclose{,} a value is called an outlier when it is more than 1.5⋅IQR1.5 \cdot \text{IQR} above Q3.Q_3\mathclose{.} Give that upper limit, Q3+1.5⋅IQR.Q_3 + 1.5 \cdot \text{IQR}\mathclose{.} Take Q1Q_1 and Q3Q_3 as the medians of the lower and upper halves.

Answer:

2

For the data set 3,28,26,26,1,12,3, \allowbreak 28, \allowbreak 26, \allowbreak 26, \allowbreak 1, \allowbreak 12\mathclose{,} give the mean.

Answer:

3

For the data set 18,12,39,36,16,38,13,3,18, \allowbreak 12, \allowbreak 39, \allowbreak 36, \allowbreak 16, \allowbreak 38, \allowbreak 13, \allowbreak 3\mathclose{,} give the interquartile range. Take Q1Q_1 and Q3Q_3 as the medians of the lower and upper halves.

Answer:

4

The data 4,5,6,17,184, \allowbreak 5, \allowbreak 6, \allowbreak 17, \allowbreak 18 have mean 1010 and median 6.6\mathclose{.} The largest value, 18,18\mathclose{,} is replaced by 100.100\mathclose{.} Give the new mean.

Answer:

5

For the data set 25,25,7,13,6,14,24,25, \allowbreak 25, \allowbreak 7, \allowbreak 13, \allowbreak 6, \allowbreak 14, \allowbreak 24\mathclose{,} give the median.

Answer:

6

For the data set 9,18,2,5,27,9, \allowbreak 18, \allowbreak 2, \allowbreak 5, \allowbreak 27\mathclose{,} give the range.

Answer:

7

For the data 57,22,60,49,52,42,29,81,73,57, \allowbreak 22, \allowbreak 60, \allowbreak 49, \allowbreak 52, \allowbreak 42, \allowbreak 29, \allowbreak 81, \allowbreak 73\mathclose{,} give the 2020th percentile. Use the rule: the ppth percentile is the value at position p(n+1)100\dfrac{p(n + 1)}{100} when the data are in order.

Answer:

8

The mean of five numbers is 20.20\mathclose{.} Four of them are 25,17,2225, 17, 22 and 13.13\mathclose{.} What is the fifth?

Answer:

9

For the data set 12,15,18,14,16,12, \allowbreak 15, \allowbreak 18, \allowbreak 14, \allowbreak 16\mathclose{,} give the standard deviation. Treat the data as a whole population, so divide by n,n\mathclose{,} the number of values.

Answer:

10

For the data 76,56,99,28,25,34,63,71,30,33,35,76, \allowbreak 56, \allowbreak 99, \allowbreak 28, \allowbreak 25, \allowbreak 34, \allowbreak 63, \allowbreak 71, \allowbreak 30, \allowbreak 33, \allowbreak 35\mathclose{,} give the third quartile Q3.Q_3\mathclose{.} Take it as the median of the upper half, leaving the middle value out of both halves.

Answer:

11

A data set has mean 4141 and median 34.34\mathclose{.} Is its distribution skewed to the left, symmetric, or skewed to the right?

Answer:

12

For the data set 20,6,9,20,23,19,13,20, \allowbreak 6, \allowbreak 9, \allowbreak 20, \allowbreak 23, \allowbreak 19, \allowbreak 13\mathclose{,} give the mode.

Answer:

13

For the data set 26,3,18,29,10,26, \allowbreak 3, \allowbreak 18, \allowbreak 29, \allowbreak 10\mathclose{,} give the variance. Treat the data as a whole population, so divide by n,n\mathclose{,} the number of values.

Answer:

14

Give the sum of squares ∑(x−xˉ)2\sum (x - \bar{x})^2 of the data 8,7,17,18,20.8, \allowbreak 7, \allowbreak 17, \allowbreak 18, \allowbreak 20\mathclose{.}

Answer:

15

For the data set 9,23,26,1,4,10,4,9, \allowbreak 23, \allowbreak 26, \allowbreak 1, \allowbreak 4, \allowbreak 10, \allowbreak 4\mathclose{,} give the mean.

Answer:

16

For the data set 16,27,10,35,29,25,3,18,16, \allowbreak 27, \allowbreak 10, \allowbreak 35, \allowbreak 29, \allowbreak 25, \allowbreak 3, \allowbreak 18\mathclose{,} give the interquartile range. Take Q1Q_1 and Q3Q_3 as the medians of the lower and upper halves.

Answer:

17

For the data 34,33,71,48,32,30,45,47,34, \allowbreak 33, \allowbreak 71, \allowbreak 48, \allowbreak 32, \allowbreak 30, \allowbreak 45, \allowbreak 47\mathclose{,} a value is called an outlier when it is more than 1.5⋅IQR1.5 \cdot \text{IQR} above Q3.Q_3\mathclose{.} Give that upper limit, Q3+1.5⋅IQR.Q_3 + 1.5 \cdot \text{IQR}\mathclose{.} Take Q1Q_1 and Q3Q_3 as the medians of the lower and upper halves.

Answer:

18

For the data set 17,29,30,30,10,11,17,17, \allowbreak 29, \allowbreak 30, \allowbreak 30, \allowbreak 10, \allowbreak 11, \allowbreak 17\mathclose{,} give the median.

Answer:

19

For the data set 11,15,20,20,11,11, \allowbreak 15, \allowbreak 20, \allowbreak 20, \allowbreak 11\mathclose{,} give the range.

Answer:

20

The data 3,6,12,14,153, \allowbreak 6, \allowbreak 12, \allowbreak 14, \allowbreak 15 have mean 1010 and median 12.12\mathclose{.} The largest value, 15,15\mathclose{,} is replaced by 100.100\mathclose{.} Give the new mean.

Answer:

21

For the data 76,42,65,50,87,12,34,41,56,76, \allowbreak 42, \allowbreak 65, \allowbreak 50, \allowbreak 87, \allowbreak 12, \allowbreak 34, \allowbreak 41, \allowbreak 56\mathclose{,} give the 2020th percentile. Use the rule: the ppth percentile is the value at position p(n+1)100\dfrac{p(n + 1)}{100} when the data are in order.

Answer:

22

The mean of five numbers is 17.17\mathclose{.} Four of them are 21,24,1621, 24, 16 and 13.13\mathclose{.} What is the fifth?

Answer:

23

For the data set 12,10,3,10,6,7,12, \allowbreak 10, \allowbreak 3, \allowbreak 10, \allowbreak 6, \allowbreak 7\mathclose{,} give the standard deviation. Treat the data as a whole population, so divide by n,n\mathclose{,} the number of values.

Answer:

24

For the data 96,88,12,43,66,20,29,71,80,75,16,96, \allowbreak 88, \allowbreak 12, \allowbreak 43, \allowbreak 66, \allowbreak 20, \allowbreak 29, \allowbreak 71, \allowbreak 80, \allowbreak 75, \allowbreak 16\mathclose{,} give the third quartile Q3.Q_3\mathclose{.} Take it as the median of the upper half, leaving the middle value out of both halves.

Answer:

25

For the data set 24,13,6,6,11,24, \allowbreak 13, \allowbreak 6, \allowbreak 6, \allowbreak 11\mathclose{,} give the mode.

Answer:

26

For the data set 12,17,20,10,3,12, \allowbreak 17, \allowbreak 20, \allowbreak 10, \allowbreak 3\mathclose{,} give the variance. Treat the data as a whole population, so divide by n,n\mathclose{,} the number of values.

Answer:

27

A data set has mean 5757 and median 51.51\mathclose{.} Is its distribution skewed to the left, symmetric, or skewed to the right?

Answer:

28

For the data set 28,16,10,15,22,30,28, \allowbreak 16, \allowbreak 10, \allowbreak 15, \allowbreak 22, \allowbreak 30\mathclose{,} give the mean.

Answer:

29

For the data set 33,14,5,15,39,32,26,28,33, \allowbreak 14, \allowbreak 5, \allowbreak 15, \allowbreak 39, \allowbreak 32, \allowbreak 26, \allowbreak 28\mathclose{,} give the interquartile range. Take Q1Q_1 and Q3Q_3 as the medians of the lower and upper halves.

Answer:

30

Give the sum of squares ∑(x−xˉ)2\sum (x - \bar{x})^2 of the data 3,7,12,5,13.3, \allowbreak 7, \allowbreak 12, \allowbreak 5, \allowbreak 13\mathclose{.}

Answer:

Answers

Every answer below was re-derived independently before this page was built.

1

Answer: 113113

In order: 22,23,33,35,41,53,71,79.22, \allowbreak 23, \allowbreak 33, \allowbreak 35, \allowbreak 41, \allowbreak 53, \allowbreak 71, \allowbreak 79\mathclose{.} Q1=28Q_1 = 28 and Q3=62,Q_3 = 62\mathclose{,} so IQR=34\text{IQR} = 34 and the limit is 62+1.5⋅34=113.62 + 1.5 \cdot 34 = 113\mathclose{.}

Common mistakes, and the answer each one gives:

  • Added 1.5⋅IQR1.5 \cdot \text{IQR} to the median instead of to Q3Q_3 → 8989
  • Added the IQR without the factor 1.51.5 → 9696
2

Answer: 1616

The mean is the total divided by how many values there are: 966=16.\dfrac{96}{6} = 16\mathclose{.}

Common mistakes, and the answer each one gives:

  • Gave the median instead of the mean → 1919
3

Answer: 24.524.5

In order: 3,12,13,16,18,36,38,39.3, \allowbreak 12, \allowbreak 13, \allowbreak 16, \allowbreak 18, \allowbreak 36, \allowbreak 38, \allowbreak 39\mathclose{.} The lower half is the first four, with median Q1=12.5;Q_1 = 12.5\mathclose{;} the upper half the last four, with Q3=37.Q_3 = 37\mathclose{.} So the IQR is 37−12.5=24.5.37 - 12.5 = 24.5\mathclose{.}

Common mistakes, and the answer each one gives:

  • Gave the range instead → 3636
  • Took the quartiles from the list AS WRITTEN, without sorting it first → 00
4

Answer: 26.426.4

The sum grows by 100−18=82,100 - 18 = 82\mathclose{,} from 5050 to 132,132\mathclose{,} so the mean becomes 1325=26.4\dfrac{132}{5} = 26.4 — while the median stays 6.6\mathclose{.} The mean is sensitive to an outlier; the median is not.

Common mistakes, and the answer each one gives:

  • Gave the new median, which has not moved → 66
  • Added the whole change to the old mean, without dividing by 5 → 9292
5

Answer: 1414

Put the values in order first — 6,7,13,14,24,25,256, \allowbreak 7, \allowbreak 13, \allowbreak 14, \allowbreak 24, \allowbreak 25, \allowbreak 25 — then take the middle one, 14.14\mathclose{.}

Common mistakes, and the answer each one gives:

  • Gave the mean instead of the median → 1147\frac{114}{7}
  • Took the middle of the list AS WRITTEN, without sorting it first → 1313
6

Answer: 2525

The range is the largest value minus the smallest: 27−2=25.27 - 2 = 25\mathclose{.}

Common mistakes, and the answer each one gives:

  • Subtracted the first value from the last AS WRITTEN → 1818
7

Answer: 2929

In order: 22,29,42,49,52,57,60,73,81.22, \allowbreak 29, \allowbreak 42, \allowbreak 49, \allowbreak 52, \allowbreak 57, \allowbreak 60, \allowbreak 73, \allowbreak 81\mathclose{.} With n=9n = 9 the position is 20⋅10100=2,\dfrac{20 \cdot 10}{100} = 2\mathclose{,} and the 2nd value is 29.29\mathclose{.}

Common mistakes, and the answer each one gives:

  • Counted the position from the largest value → 7373
  • Did not put the data in order first → 2222
8

Answer: 2323

A mean of 2020 over five numbers means a total of 5×20=100.5 \times 20 = 100\mathclose{.} The four add to 77,77\mathclose{,} so the fifth is 100−77=23.100 - 77 = 23\mathclose{.}

Common mistakes, and the answer each one gives:

  • Gave the mean of the four numbers given → 19.2519.25
  • Used 4×204 \times 20 as the total → 33
9

Answer: 22

The mean is 15.15\mathclose{.} The squared distances from it add to 20,20\mathclose{,} so the variance is 205=4\dfrac{20}{5} = 4 and the standard deviation is 4=2.\sqrt{4} = 2\mathclose{.}

Common mistakes, and the answer each one gives:

  • Gave the variance instead of the standard deviation → 44
  • Divided by n−1,n - 1\mathclose{,} the formula for a sample → 5\sqrt{5}
10

Answer: 7171

In order: 25,28,30,33,34,35,56,63,71,76,99.25, \allowbreak 28, \allowbreak 30, \allowbreak 33, \allowbreak 34, \allowbreak 35, \allowbreak 56, \allowbreak 63, \allowbreak 71, \allowbreak 76, \allowbreak 99\mathclose{.} The median is the 6th value, 35;35\mathclose{;} the upper half is 56,63,71,76,99,56, \allowbreak 63, \allowbreak 71, \allowbreak 76, \allowbreak 99\mathclose{,} and its median is 71.71\mathclose{.}

Common mistakes, and the answer each one gives:

  • Kept the median in the half → 6767
  • Did not put the data in order first → 3535
11

Answer: skewed to the right\text{skewed to the right}

The mean is pulled towards the long tail; the median is not. The mean is larger than the median, so the tail is on the right: skewed to the right.

Common mistakes, and the answer each one gives:

  • Took the skew to be on the side where most of the data lie → skewed to the left\text{skewed to the left}
12

Answer: 2020

The mode is the value that occurs most often: 2020 occurs 22 times and every other value once.

Common mistakes, and the answer each one gives:

  • Gave how many times it occurs instead of the value → 22
  • Gave the median instead → 1919
13

Answer: 94.1694.16

The mean is 17.2.17.2\mathclose{.} Square each value's distance from it, add those up — that gives 470.8470.8 — and divide by 5:5\mathclose{:} 94.16.94.16\mathclose{.}

Common mistakes, and the answer each one gives:

  • Divided by n−1,n - 1\mathclose{,} the formula for a sample → 117.7117.7
  • Added the distances from the mean without squaring them → 8.568.56
14

Answer: 146146

The mean is xˉ=14.\bar{x} = 14\mathclose{.} The deviations from it are −6,−7,3,4,6;-6, \allowbreak -7, \allowbreak 3, \allowbreak 4, \allowbreak 6\mathclose{;} squared and added: 36+49+9+16+36=146.36 + 49 + 9 + 16 + 36 = 146\mathclose{.}

Common mistakes, and the answer each one gives:

  • Divided by n,n\mathclose{,} which gives the variance instead → 29.229.2
  • Added the deviations without squaring them → 00
15

Answer: 1111

The mean is the total divided by how many values there are: 777=11.\dfrac{77}{7} = 11\mathclose{.}

Common mistakes, and the answer each one gives:

  • Gave the median instead of the mean → 99
16

Answer: 1515

In order: 3,10,16,18,25,27,29,35.3, \allowbreak 10, \allowbreak 16, \allowbreak 18, \allowbreak 25, \allowbreak 27, \allowbreak 29, \allowbreak 35\mathclose{.} The lower half is the first four, with median Q1=13;Q_1 = 13\mathclose{;} the upper half the last four, with Q3=28.Q_3 = 28\mathclose{.} So the IQR is 28−13=15.28 - 13 = 15\mathclose{.}

Common mistakes, and the answer each one gives:

  • Gave the range instead → 3232
  • Took the quartiles from the list AS WRITTEN, without sorting it first → −4.5-4.5
17

Answer: 7070

In order: 30,32,33,34,45,47,48,71.30, \allowbreak 32, \allowbreak 33, \allowbreak 34, \allowbreak 45, \allowbreak 47, \allowbreak 48, \allowbreak 71\mathclose{.} Q1=32.5Q_1 = 32.5 and Q3=47.5,Q_3 = 47.5\mathclose{,} so IQR=15\text{IQR} = 15 and the limit is 47.5+1.5⋅15=70.47.5 + 1.5 \cdot 15 = 70\mathclose{.}

Common mistakes, and the answer each one gives:

  • Added 1.5⋅IQR1.5 \cdot \text{IQR} to the median instead of to Q3Q_3 → 6262
  • Added the IQR without the factor 1.51.5 → 62.562.5
18

Answer: 1717

Put the values in order first — 10,11,17,17,29,30,3010, \allowbreak 11, \allowbreak 17, \allowbreak 17, \allowbreak 29, \allowbreak 30, \allowbreak 30 — then take the middle one, 17.17\mathclose{.}

Common mistakes, and the answer each one gives:

  • Gave the mean instead of the median → 1447\frac{144}{7}
  • Took the middle of the list AS WRITTEN, without sorting it first → 3030
19

Answer: 99

The range is the largest value minus the smallest: 20−11=9.20 - 11 = 9\mathclose{.}

Common mistakes, and the answer each one gives:

  • Subtracted the first value from the last AS WRITTEN → 00
20

Answer: 2727

The sum grows by 100−15=85,100 - 15 = 85\mathclose{,} from 5050 to 135,135\mathclose{,} so the mean becomes 1355=27\dfrac{135}{5} = 27 — while the median stays 12.12\mathclose{.} The mean is sensitive to an outlier; the median is not.

Common mistakes, and the answer each one gives:

  • Gave the new median, which has not moved → 1212
  • Added the whole change to the old mean, without dividing by 5 → 9595
21

Answer: 3434

In order: 12,34,41,42,50,56,65,76,87.12, \allowbreak 34, \allowbreak 41, \allowbreak 42, \allowbreak 50, \allowbreak 56, \allowbreak 65, \allowbreak 76, \allowbreak 87\mathclose{.} With n=9n = 9 the position is 20⋅10100=2,\dfrac{20 \cdot 10}{100} = 2\mathclose{,} and the 2nd value is 34.34\mathclose{.}

Common mistakes, and the answer each one gives:

  • Counted the position from the largest value → 7676
  • Did not put the data in order first → 4242
22

Answer: 1111

A mean of 1717 over five numbers means a total of 5×17=85.5 \times 17 = 85\mathclose{.} The four add to 74,74\mathclose{,} so the fifth is 85−74=11.85 - 74 = 11\mathclose{.}

Common mistakes, and the answer each one gives:

  • Gave the mean of the four numbers given → 18.518.5
  • Used 4×174 \times 17 as the total → −6-6
23

Answer: 33

The mean is 8.8\mathclose{.} The squared distances from it add to 54,54\mathclose{,} so the variance is 546=9\dfrac{54}{6} = 9 and the standard deviation is 9=3.\sqrt{9} = 3\mathclose{.}

Common mistakes, and the answer each one gives:

  • Gave the variance instead of the standard deviation → 99
  • Divided by n−1,n - 1\mathclose{,} the formula for a sample → 3305\frac{3 \sqrt{30}}{5}
24

Answer: 8080

In order: 12,16,20,29,43,66,71,75,80,88,96.12, \allowbreak 16, \allowbreak 20, \allowbreak 29, \allowbreak 43, \allowbreak 66, \allowbreak 71, \allowbreak 75, \allowbreak 80, \allowbreak 88, \allowbreak 96\mathclose{.} The median is the 6th value, 66;66\mathclose{;} the upper half is 71,75,80,88,96,71, \allowbreak 75, \allowbreak 80, \allowbreak 88, \allowbreak 96\mathclose{,} and its median is 80.80\mathclose{.}

Common mistakes, and the answer each one gives:

  • Kept the median in the half → 77.577.5
  • Did not put the data in order first → 7171
25

Answer: 66

The mode is the value that occurs most often: 66 occurs 22 times and every other value once.

Common mistakes, and the answer each one gives:

  • Gave how many times it occurs instead of the value → 22
  • Gave the median instead → 1111
26

Answer: 34.6434.64

The mean is 12.4.12.4\mathclose{.} Square each value's distance from it, add those up — that gives 173.2173.2 — and divide by 5:5\mathclose{:} 34.64.34.64\mathclose{.}

Common mistakes, and the answer each one gives:

  • Divided by n−1,n - 1\mathclose{,} the formula for a sample → 43.343.3
  • Added the distances from the mean without squaring them → 4.884.88
27

Answer: skewed to the right\text{skewed to the right}

The mean is pulled towards the long tail; the median is not. The mean is larger than the median, so the tail is on the right: skewed to the right.

Common mistakes, and the answer each one gives:

  • Took the skew to be on the side where most of the data lie → skewed to the left\text{skewed to the left}
28

Answer: 1216\frac{121}{6}

The mean is the total divided by how many values there are: 1216.\dfrac{121}{6}\mathclose{.}

Common mistakes, and the answer each one gives:

  • Gave the median instead of the mean → 1919
29

Answer: 1818

In order: 5,14,15,26,28,32,33,39.5, \allowbreak 14, \allowbreak 15, \allowbreak 26, \allowbreak 28, \allowbreak 32, \allowbreak 33, \allowbreak 39\mathclose{.} The lower half is the first four, with median Q1=14.5;Q_1 = 14.5\mathclose{;} the upper half the last four, with Q3=32.5.Q_3 = 32.5\mathclose{.} So the IQR is 32.5−14.5=18.32.5 - 14.5 = 18\mathclose{.}

Common mistakes, and the answer each one gives:

  • Gave the range instead → 3434
  • Took the quartiles from the list AS WRITTEN, without sorting it first → 19.519.5
30

Answer: 7676

The mean is xˉ=8.\bar{x} = 8\mathclose{.} The deviations from it are −5,−1,4,−3,5;-5, \allowbreak -1, \allowbreak 4, \allowbreak -3, \allowbreak 5\mathclose{;} squared and added: 25+1+16+9+25=76.25 + 1 + 16 + 9 + 25 = 76\mathclose{.}

Common mistakes, and the answer each one gives:

  • Divided by n,n\mathclose{,} which gives the variance instead → 15.215.2
  • Added the deviations without squaring them → 00