Worksheet: Sample and Population Standard Deviation

In this worksheet, we will practice interpreting the variation in data by learning about sample and population standard deviation.

Q1:

What is the denominator in the calculation of population standard deviation for a data set with 𝑛 items?

  • A 𝑛 1
  • B 𝑛 + 2
  • C 𝑛
  • D 𝑛 + 1
  • E 𝑛 2

Q2:

What is the denominator in the calculation of sample standard deviation for a data set with 𝑛 items?

  • A 𝑛 2
  • B 𝑛 1
  • C 𝑛 + 2
  • D 𝑛 + 1
  • E 𝑛

Q3:

Calculate, to the nearest hundredth, the sample standard deviation for each of the shown data sets.

Data Set 1 25 22 28 51 26 28 29 32
Data Set 2 21 27 19 26 24 23 28 25
  • Adata set 1: 𝑠=8.36, data set 2: 𝑠=2.85
  • Bdata set 1: 𝑠=8.94, data set 2: 𝑠=3.04
  • Cdata set 1: 𝑠=8.94, data set 2: 𝑠=2.85
  • Ddata set 1: 𝑠=69.86, data set 2: 𝑠=8.11
  • Edata set 1: 𝑠=79.84, data set 2: 𝑠=9.27

Which of these statements helps explain the large difference between the sample standard deviations for the two data sets?

  • AThe lowest value is in data set 2, so this makes the standard deviation smaller.
  • BThe single large value of 51 in data set 1 causes the standard deviation to increase significantly.
  • CThe median of data set 1 is greater, which explains why the standard deviation is greater.
  • DThe mode of data set 1 value of 28 causes the standard deviation to increase significantly.
  • EThe mean of data set 1 is greater, which explains why the standard deviation is greater.

Q4:

The data set shown contains a potentially outlying value of 32. What effect would removing this value have on the standard deviation?

12 15 13 11 32 13 9 19
  • AThe standard deviation would decrease.
  • BThe standard deviation would increase.
  • CThe standard deviation would not change.

Q5:

Without calculating the exact standard deviations of the following data sets, determine which of them has the lowest standard deviation.

  • A 75, 75, 75, 75, 75, 1,500
  • B10, 20, 30, 40, 50, 60
  • C41, 41, 41, 41, 41, 42
  • D35, 38, 42, 48, 48, 48
  • E100, 200, 300, 400, 500, 600

Q6:

Using the data given in the table, calculate the standard deviation of the number of children. If necessary, give your answer to three decimal places.

Number of Children 1 2 3 4 5
Number of Families 15 26 3 28 14

Q7:

In general, what does a larger value of σ mean?

  • AThe data is less spread out.
  • BThe mean of the data is smaller.
  • CThe data is more spread out.
  • DThe mean of the data is larger.
  • EThe median is larger than the mean.

Q8:

Data Set 1 1 7 10 9 8 2 2 1
Data Set 2 3 7 8 6 5 4 4 3

Calculate, to the nearest hundredth, the sample standard deviation for each data set.

  • Adata set 1: 14.86, data set 2:3.43
  • Bdata set 1: 3.85, data set 2: 1.85
  • Cdata set 1: 3.61, data set 2: 1.73
  • Ddata set 1: 1.85, data set 2: 3.85
  • Edata set 1: 13.00, data set 2:3.00

What do these values reveal about the two data sets?

  • AThe spread of the two data sets is very similar.
  • BData set 2 is more widely spread than data set 1.
  • CData set 2 is more variable than data set 1.
  • DData set 1 is more widely spread than data set 2.
  • EThe distributions of the two data sets are very similar.

Q9:

The following data represents the morning temperatures in degrees Fahrenheit and the rainfall in millimeters for all the Canadian cities east of Toronto in July.

1 3 . 5 F 1 6 . 1 F 1 2 . 5 F 1 5 . 5 F 1 8 . 7 F 1 8 . 7 F 1 4 . 7 F 1 8 . 3 F 1 5 . 8 F
1 2 F 1 7 . 5 F 1 8 . 2 F 1 4 F 1 4 . 1 F 1 8 . 4 F 1 4 . 8 F 1 1 . 5 F 1 8 . 1 F
82.5 mm 47.6 mm 50.7 mm 95.6 mm 70.6 mm 66.3 mm 36.6 mm 48.3 mm 56 mm
64.9 mm 69.2 mm 46.2 mm 101.3 mm 86.8 mm 82.4 mm 106.1 mm 94.7 mm 57.2 mm

By considering the standard deviation, determine which data set is more variable. State the standard deviation of this data set to three decimal places.

  • Aprecipitation, 20.504
  • Btemperature, 21.099
  • Ctemperature, 2.371
  • Dprecipitation, 21.099
  • Etemperature, 20.504

Q10:

Using the data given in the table, calculate the standard deviation of the ages. Give your answer to three decimal places.

Ages (years) 5 8 10 12 15
Number of Students 1 9 8 9 8

Q11:

Calculate the standard deviation of this frequency distribution of the ages of a group of people. Round your answer to the nearest thousandth.

Age 25–34 35–44 45–54 55–64 65–74
Frequency 35 45 20 30 20

Q12:

The weights of 150 students are recorded in the frequency table.

Weight (𝑤)kg 3 0 𝑤 < 4 0 4 0 𝑤 < 5 0 5 0 𝑤 < 6 0 6 0 𝑤 < 7 0 7 0 𝑤 < 8 0
Number of Students 45 45 25 10 25

Calculate the standard deviation, rounding to the nearest thousandth.

Q13:

The given table is the frequency distribution for a number of defective units found in 80 boxes of manufactured units.

Number of defective units 0 1 2 3 4 5
Number of boxes 8 4 15 20 19 14

Find the standard deviation of the number of the defective units, and round the result to the nearest thousandth.

Q14:

The scores achieved by students in an exam are recorded in the frequency table. Calculate the standard deviation, rounding the result to the nearest thousandth.

Score ()𝑠 0 𝑠 < 4 4 𝑠 < 8 8 𝑠 < 1 2 1 2 𝑠 < 1 6 1 6 𝑠 < 2 0
Frequency 2 2 8 6 6

Q15:

Using the data given in the table, calculate the standard deviation mark. If necessary, give your answer to three decimal places.

Sets of Marks 0–9 10–19 20–29 30–39 40–49
Number of Students 18 27 22 19 22

Q16:

The scores of 76 students in an exam are recorded in the frequency table.

Score ()𝑚 0 𝑚 < 2 2 𝑚 < 4 4 𝑚 < 6 6 𝑚 < 8 8 𝑚 < 1 0
Frequency 13 19 15 13 16

Calculate the standard deviation, rounding the result to the nearest thousandth if necessary.

Q17:

Find, to two decimal places, the standard deviation of the following scores obtained in a quiz by 85 students.

Score 0–10 10–20 20–30 30–40 40–50
Frequency 11 18 29 5 22

Q18:

William is playing a gambling game where he will either win money, lose money, or break even, as shown in the table.

Outcome $339 $0 $319 $271
Probability 0.24 0.38 0.22 0.16

Let 𝑥 be the outcome for this game. Calculate, to the nearest hundredth, the standard deviation of 𝑥.

Q19:

The table shows the greatest distances thrown by the finalists in the discus competition of the 2016 Rio Olympics for both men and women.

Distance (m)
Men Women
Gold 68.37 69.21
Silver 67.55 66.73
Bronze 67.05 65.34
4 66.58 64.90
5 65.10 64.37
6 64.95 63.13
7 64.50 63.11
8 63.72 63.06

Calculate, to the nearest hundredth of a meter, the population standard deviation of the distances thrown by the men and the distances thrown by the women.

  • Amen: 2.00 m, women: 1.53 m
  • Bmen: 1.64 m, women: 2.14 m
  • Cmen: 1.64 m, women: 2.00 m
  • Dmen: 1.53 m, women: 2.00 m
  • Emen: 1.53 m, women: 2.14 m

What is the correct interpretation of the population standard deviations?

  • AOn average, women threw farther than men.
  • BThe distributions of the two data sets are very similar.
  • CThere was a greater variability in the distances thrown by women.
  • DThere was a greater variability in the distances thrown by men.
  • EOn average, men threw farther than women.

Q20:

The following data represents the morning temperatures in degrees Fahrenheit and the rainfall in millimeters for all the Canadian cities east of Toronto in July.

1 7 . 5 F 1 3 . 5 F 1 7 . 1 F 1 1 . 7 F 1 4 . 3 F 1 0 . 8 F 1 7 . 9 F 1 2 . 4 F 1 2 . 6 F
1 7 . 8 F 1 8 . 5 F 1 1 . 4 F 1 0 . 5 F 1 8 . 2 F 1 0 . 2 F 1 8 F 1 2 . 2 F 1 5 . 2 F
92.3 mm 72.7 mm 35.7 mm 108.1 mm 46.4 mm 37.5 mm 41.8 mm 36.3 mm 54.4 mm
85.9 mm 58.5 mm 63.6 mm 97.1 mm 75.8 mm 76.9 mm 50.8 mm 86.6 mm 69.4 mm

By considering the standard deviation, determine which data set is more variable. State the standard deviation of this data set to three decimal places.

  • Aprecipitation, 21.772
  • Btemperature, 22.404
  • Ctemperature, 2.986
  • Dprecipitation, 22.404
  • Etemperature, 21.772

Q21:

What does high sample variability mean?

  • AThe means of the samples have large values.
  • BA large variety of methods have been used to generate the samples.
  • CGiven multiple samples from the same population, we should expect the sample statistics, for example, the calculated means, to have largely different values.
  • DThere is a large number of samples.

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