# 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 items?

- A
- B
- C
- D
- E

**Q2: **

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

- A
- B
- C
- D
- E

**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: , data set 2:
- Bdata set 1: , data set 2:
- Cdata set 1: , data set 2:
- Ddata set 1: , data set 2:
- Edata set 1: , data set 2:

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.

**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.

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 | |||||
---|---|---|---|---|---|

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 | |||||
---|---|---|---|---|---|

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 | |||||
---|---|---|---|---|---|

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

**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.

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.