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Worksheet: Sample and Population Standard Deviation

Q1:

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

  • A100, 200, 300, 400, 500, 600
  • B75, 75, 75, 75, 75, 1 500
  • C10, 20, 30, 40, 50, 60
  • D41, 41, 41, 41, 41, 42
  • E35, 38, 42, 48, 48, 48

Q2:

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

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 . 3 6 , data set 2: 𝑠 = 2 . 8 5
  • Bdata set 1: 𝑠 = 7 9 . 8 4 , data set 2: 𝑠 = 9 . 2 7
  • Cdata set 1: 𝑠 = 6 9 . 8 6 , data set 2: 𝑠 = 8 . 1 1
  • Ddata set 1: 𝑠 = 8 . 9 4 , data set 2: 𝑠 = 3 . 0 4
  • Edata set 1: 𝑠 = 8 . 9 4 , data set 2: 𝑠 = 2 . 8 5

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

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

Q4:

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

Q5:

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

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

Q6:

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

Q7:

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

Q8:

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.

  • Atemperature, 20.504
  • Btemperature, 21.099
  • Cprecipitation, 21.099
  • Dprecipitation, 20.504
  • Etemperature, 2.371

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 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.

  • Atemperature, 21.772
  • Btemperature, 22.404
  • Cprecipitation, 22.404
  • Dprecipitation, 21.772
  • Etemperature, 2.986

Q10:

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

What do these values reveal about the two data sets?

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

Q11:

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

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

Let π‘₯ be the outcome for this game. Calculate, to the nearest hundredth, the standard deviation of π‘₯ .

Q12:

The weights of 175 students are recorded in the frequency table.

Weight ( 𝑀 ) k g 3 0 ≀ 𝑀 < 4 0 4 0 ≀ 𝑀 < 5 0 5 0 ≀ 𝑀 < 6 0 6 0 ≀ 𝑀 < 7 0 7 0 ≀ 𝑀 < 8 0
Number of Students 40 50 15 45 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:

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

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

Q15:

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

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

Q16:

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

  • A The mean of the data is larger.
  • B The data is less spread out.
  • C The mean of the data is smaller.
  • D The data is more spread out.
  • E The median is larger than the mean.

Q17:

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

What is the correct interpretation of the population standard deviations?

  • AThere was a greater variability in the distances thrown by women.
  • BOn average, women threw farther than men.
  • COn average, men threw farther than women.
  • DThere was a greater variability in the distances thrown by men.
  • EThe distributions of the two data sets are very similar.

Q18:

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

Q19:

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 increase.
  • BThe standard deviation would not change.
  • CThe standard deviation would decrease.

Q20:

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

Mark, 0 ≀ π‘š < 2 2 ≀ π‘š < 4 4 ≀ π‘š < 6 6 ≀ π‘š < 8 8 ≀ π‘š < 1 0
Frequency 13 19 15 13 16

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