Standard deviation (tabular method) in real world problems worksheets, solved worksheet problems or exercises with step-by-step work, formative assessment as online test, calculator and more learning resources to learn, practice, assess and master the basic math skills of **statistics**.

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1

Find the sample standard deviation of the following data. 8, 5, 2, 4, 10, 1, 7, 3, 6 and 9

2

Find the population standard deviation of the following data. 6, 8, 10, 12, 14, 16, 18, 20, 22 and 24

3

A teacher asked the students to complete 60 pages of a record notebook. Eight students have completed only 32, 35, 37, 30, 33, 36, 35 and 37 pages. Find the standard deviation of the pages yet to be completed by them.

4

Find the standard deviation of the wages of 9 workers given in ₹: 310, 290, 320, 280, 300, 290, 320, 310 and 280

5

Find the standard deviation of the first 10 natural numbers.

6

The amount of rainfall in a particular season for 6 days are given in centimeter are 17.8, 19.2, 16.3, 12.5, 12.8 and 11.4. Find its standard deviation.

7

Find the standard deviation of the average temperatures recorded over a five-day period last winter in celcius: 18, 22, 19, 25 and 12

8

Find the standard deviation of the highest temperatures recorded in eight specific states in celcius: 112, 100, 127, 120, 134, 118, 105 and 110

9

Find the standard deviation of the scores on the most recent reading test: 7.7, 7.4, 7.3 and 7.9

10

During a survey, 6 students were asked how many hours per day they study on an average? Their answers were as follows: 2, 6, 5, 3, 2 and 3. Evaluate the standard deviation.

Answers Key

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1

Find the sample standard deviation of the following data. 8, 5, 2, 4, 10, 1, 7, 3, 6 and 9

Sample Standard Deviation is 3.03

Solution

s=√∑^{n}_{i=1}(𝑥_{i} − x̄)^{2}(n-1)

s=√∑^{n}_{i=1}(𝑥_{i} − x̄)^{2}(n-1)

s = √82.5(10 - 1)

s = √82.59

s = √9.17

s=√∑

where s = Sample Standard Deviation,

𝑥_{i} = Terms Given in the Data,

x̄ = Mean,

n = Total number of Terms

𝑥

x̄ = Mean,

n = Total number of Terms

x_{i} | (𝑥_{i} − x̄) | (𝑥_{i} − x̄)^{2} |
---|---|---|

8 | 2.5 | 6.25 |

5 | -0.5 | 0.25 |

2 | -3.5 | 12.25 |

4 | -1.5 | 2.25 |

10 | 4.5 | 20.25 |

1 | -4.5 | 20.25 |

7 | 1.5 | 2.25 |

3 | -2.5 | 6.25 |

6 | 0.5 | 0.25 |

9 | 3.5 | 12.25 |

∑𝑥_{i} = 55 | ∑(𝑥_{i} − x̄)^{2} = 82.5 | |

x̄ = 55/10 = 5.5 |

s=√∑

s = √82.5(10 - 1)

s = √82.59

s = √9.17

∴ Sample Standard Deviation, s = 3.03

2

Find the population standard deviation of the following data. 6, 8, 10, 12, 14, 16, 18, 20, 22 and 24

Population Standard Deviation is 5.74

Solution

σ=√∑^{n}_{i=1}(𝑥_{i} − μ)^{2}N

σ=√∑^{n}_{i=1}(𝑥_{i} − μ)^{2}N

σ = √33010

σ = √33

σ=√∑

where σ = Population Standard Deviation,

𝑥_{i} = Terms Given in the Data,

μ = Mean,

N = Total number of Terms

𝑥

μ = Mean,

N = Total number of Terms

x_{i} | (𝑥_{i} − μ) | (𝑥_{i} − μ)^{2} |
---|---|---|

6 | -9 | 81 |

8 | -7 | 49 |

10 | -5 | 25 |

12 | -3 | 9 |

14 | -1 | 1 |

16 | 1 | 1 |

18 | 3 | 9 |

20 | 5 | 25 |

22 | 7 | 49 |

24 | 9 | 81 |

∑𝑥_{i} = 150 | ∑(𝑥_{i} − μ)^{2} = 330 | |

x̄ = 150/10 = 15 |

σ=√∑

σ = √33010

σ = √33

∴ Population Standard Deviation, σ = 5.74

3

A teacher asked the students to complete 60 pages of a record notebook. Eight students have completed only 32, 35, 37, 30, 33, 36, 35 and 37 pages. Find the standard deviation of the pages yet to be completed by them.

Standard Deviation is 2.34

Solution

σ=√∑^{n}_{i=1}(𝑥_{i} − μ)^{2}N

σ=√∑^{n}_{i=1}(𝑥_{i} − μ)^{2}N

σ = √43.8788

σ = √5.485

σ=√∑

where σ = Population Standard Deviation,

𝑥_{i} = Terms Given in the Data,

μ = Mean,

N = Total number of Terms

𝑥

μ = Mean,

N = Total number of Terms

x_{i} | (𝑥_{i} − μ) | (𝑥_{i} − μ)^{2} |
---|---|---|

32 | -2.38 | 5.664 |

35 | 0.62 | 0.384 |

37 | 2.62 | 6.864 |

30 | -4.38 | 19.184 |

33 | -1.38 | 1.904 |

36 | 1.62 | 2.624 |

35 | 0.62 | 0.384 |

37 | 2.62 | 6.864 |

∑𝑥_{i} = 275 | ∑(𝑥_{i} − μ)^{2} = 43.878 | |

x̄ = 275/8 = 34.38 |

σ=√∑

σ = √43.8788

σ = √5.485

∴ Population Standard Deviation, σ = 2.34

4

Find the standard deviation of the wages of 9 workers given in ₹: 310, 290, 320, 280, 300, 290, 320, 310 and 280

Standard Deviation is 14.91

Solution

σ=√∑^{n}_{i=1}(𝑥_{i} − μ)^{2}N

σ=√∑^{n}_{i=1}(𝑥_{i} − μ)^{2}N

σ = √20009

σ = √222.222

σ=√∑

where σ = Population Standard Deviation,

𝑥_{i} = Terms Given in the Data,

μ = Mean,

N = Total number of Terms

𝑥

μ = Mean,

N = Total number of Terms

x_{i} | (𝑥_{i} − μ) | (𝑥_{i} − μ)^{2} |
---|---|---|

310 | 10 | 100 |

290 | -10 | 100 |

320 | 20 | 400 |

280 | -20 | 400 |

300 | 0 | 0 |

290 | -10 | 100 |

320 | 20 | 400 |

310 | 10 | 100 |

280 | -20 | 400 |

∑𝑥_{i} = 2700 | ∑(𝑥_{i} − μ)^{2} = 2000 | |

x̄ = 2700/9 = 300 |

σ=√∑

σ = √20009

σ = √222.222

∴ Population Standard Deviation, σ = 14.91

5

Find the standard deviation of the first 10 natural numbers.

Standard Deviation is 2.87

Solution

σ=√∑^{n}_{i=1}(𝑥_{i} − μ)^{2}N

where σ = Population Standard Deviation,

𝑥_{i} = Terms Given in the Data,

μ = Mean,

N = Total number of Terms

σ=√∑^{n}_{i=1}(𝑥_{i} − μ)^{2}N

σ = √82.510

σ = √8.25

σ=√∑

𝑥

μ = Mean,

N = Total number of Terms

x_{i} | (𝑥_{i} − μ) | (𝑥_{i} − μ)^{2} |
---|---|---|

1 | -4.5 | 20.25 |

2 | -3.5 | 12.25 |

3 | -2.5 | 6.25 |

4 | -1.5 | 2.25 |

5 | -0.5 | 0.25 |

6 | 0.5 | 0.25 |

7 | 1.5 | 2.25 |

8 | 2.5 | 6.25 |

9 | 3.5 | 12.25 |

10 | 4.5 | 20.25 |

∑𝑥_{i} = 55 | ∑(𝑥_{i} − μ)^{2} = 82.5 | |

x̄ = 55/10 = 5.5 |

σ=√∑

σ = √82.510

σ = √8.25

∴ Population Standard Deviation, σ = 2.87

6

The amount of rainfall in a particular season for 6 days are given in centimeter are 17.8, 19.2, 16.3, 12.5, 12.8 and 11.4. Find its standard deviation.

Standard Deviation is 2.92

Solution

σ=√∑^{n}_{i=1}(𝑥_{i} − μ)^{2}N

where σ = Population Standard Deviation,

𝑥_{i} = Terms Given in the Data,

μ = Mean,

N = Total number of Terms

σ=√∑^{n}_{i=1}(𝑥_{i} − μ)^{2}N

σ = √51.226

σ = √8.537

σ=√∑

𝑥

μ = Mean,

N = Total number of Terms

x_{i} | (𝑥_{i} − μ) | (𝑥_{i} − μ)^{2} |
---|---|---|

17.8 | 2.8 | 7.84 |

19.2 | 4.2 | 17.64 |

16.3 | 1.3 | 1.69 |

12.5 | -2.5 | 6.25 |

12.8 | -2.2 | 4.84 |

11.4 | -3.6 | 12.96 |

∑𝑥_{i} = 90 | ∑(𝑥_{i} − μ)^{2} = 51.22 | |

x̄ = 90/6 = 15 |

σ=√∑

σ = √51.226

σ = √8.537

∴ Population Standard Deviation, σ = 2.92

7

Find the standard deviation of the average temperatures recorded over a five-day period last winter in celcius: 18, 22, 19, 25 and 12

Standard Deviation is 4.87

Solution

s=√∑^{n}_{i=1}(𝑥_{i} − x̄)^{2}(n-1)

s=√∑^{n}_{i=1}(𝑥_{i} − x̄)^{2}(n-1)

s = √94.8(5 - 1)

s = √94.84

s = √23.7

s=√∑

where s = Sample Standard Deviation,

𝑥_{i} = Terms Given in the Data,

x̄ = Mean,

n = Total number of Terms

𝑥

x̄ = Mean,

n = Total number of Terms

x_{i} | (𝑥_{i} − x̄) | (𝑥_{i} − x̄)^{2} |
---|---|---|

18 | -1.2 | 1.44 |

22 | 2.8 | 7.84 |

19 | -0.2 | 0.04 |

25 | 5.8 | 33.64 |

12 | -7.2 | 51.84 |

∑𝑥_{i} = 96 | ∑(𝑥_{i} − x̄)^{2} = 94.8 | |

x̄ = 96/5 = 19.2 |

s=√∑

s = √94.8(5 - 1)

s = √94.84

s = √23.7

∴ Sample Standard Deviation, s = 4.87

8

Find the standard deviation of the highest temperatures recorded in eight specific states in celcius: 112, 100, 127, 120, 134, 118, 105 and 110

Standard Deviation is 11.3

Solution

s=√∑^{n}_{i=1}(𝑥_{i} − x̄)^{2}(n-1)

s=√∑^{n}_{i=1}(𝑥_{i} − x̄)^{2}(n-1)

s = √893.504(8 - 1)

s = √893.5047

s = √127.64

s=√∑

where s = Sample Standard Deviation,

𝑥_{i} = Terms Given in the Data,

x̄ = Mean,

n = Total number of Terms

𝑥

x̄ = Mean,

n = Total number of Terms

x_{i} | (𝑥_{i} − x̄) | (𝑥_{i} − x̄)^{2} |
---|---|---|

112 | -3.75 | 14.063 |

100 | -15.75 | 248.063 |

127 | 11.25 | 126.563 |

120 | 4.25 | 18.063 |

134 | 18.25 | 333.063 |

118 | 2.25 | 5.063 |

105 | -10.75 | 115.563 |

110 | -5.75 | 33.063 |

∑𝑥_{i} = 926 | ∑(𝑥_{i} − x̄)^{2} = 893.504 | |

x̄ = 926/8 = 115.75 |

s=√∑

s = √893.504(8 - 1)

s = √893.5047

s = √127.64

∴ Sample Standard Deviation, s = 11.3

9

Find the standard deviation of the scores on the most recent reading test: 7.7, 7.4, 7.3 and 7.9

Standard Deviation is 0.28

Solution

s=√∑^{n}_{i=1}(𝑥_{i} − x̄)^{2}(n-1)

where s = Sample Standard Deviation,

𝑥_{i} = Terms Given in the Data,

x̄ = Mean,

n = Total number of Terms

s=√∑^{n}_{i=1}(𝑥_{i} − x̄)^{2}(n-1)

s = √0.229(4 - 1)

s = √0.2293

s = √0.08

s=√∑

𝑥

x̄ = Mean,

n = Total number of Terms

x_{i} | (𝑥_{i} − x̄) | (𝑥_{i} − x̄)^{2} |
---|---|---|

7.7 | 0.125 | 0.016 |

7.4 | -0.175 | 0.031 |

7.3 | -0.275 | 0.076 |

7.9 | 0.325 | 0.106 |

∑𝑥_{i} = 30.3 | ∑(𝑥_{i} − x̄)^{2} = 0.229 | |

x̄ = 30.3/4 = 7.575 |

s=√∑

s = √0.229(4 - 1)

s = √0.2293

s = √0.08

∴ Sample Standard Deviation, s = 0.28

10

During a survey, 6 students were asked how many hours per day they study on an average? Their answers were as follows: 2, 6, 5, 3, 2 and 3. Evaluate the standard deviation.

Standard Deviation is 1.64

Solution

s=√∑^{n}_{i=1}(𝑥_{i} − x̄)^{2}(n-1)

where s = Sample Standard Deviation,

𝑥_{i} = Terms Given in the Data,

x̄ = Mean,

n = Total number of Terms

s=√∑^{n}_{i=1}(𝑥_{i} − x̄)^{2}(n-1)

s = √13.5(6 - 1)

s = √13.55

s = √2.7

s=√∑

𝑥

x̄ = Mean,

n = Total number of Terms

x_{i} | (𝑥_{i} − x̄) | (𝑥_{i} − x̄)^{2} |
---|---|---|

2 | -1.5 | 2.25 |

6 | 2.5 | 6.25 |

5 | 1.5 | 2.25 |

3 | -0.5 | 0.25 |

2 | -1.5 | 2.25 |

3 | -0.5 | 0.25 |

∑𝑥_{i} = 21 | ∑(𝑥_{i} − x̄)^{2} = 13.5 | |

x̄ = 21/6 = 3.5 |

s=√∑

s = √13.5(6 - 1)

s = √13.55

s = √2.7

∴ Sample Standard Deviation, s = 1.64

Online Test

Start my Challenge

Find the sample standard deviation of the following data. 4, 25, 14, 3, 17, 2, 8, 12, 11 and 23

Find the standard deviation of the first 6 natural numbers.

Find the population standard deviation of the following data. 25, 21, 19, 5, 7, 1, 20, 6, 11 and 3