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Chapter 3: Problem 32

The following data set belongs to a population: $$ \begin{array}{cccccccc} 5 & -7 & 2 & 0 & -9 & 16 & 10 & 7 \end{array} $$ Calculate the range, variance, and standard deviation.

Short Answer

Expert verified
The range of the data set is 25, the variance is 72.5 and the standard deviation is 8.52.

Step by step solution

01

Calculate the Range

First, identify the largest and smallest numbers in the dataset. In this case, the largest number is 16 and the smallest is -9. Calculate the range by subtracting the smallest number from the largest one. The range would be \(16 - (-9) = 25\)
02

Calculate the Variance

Variance is the average of the squared differences from the mean. First, calculate the mean (average) of the dataset by summing up all of the numbers and dividing by the number of figures. That gives \((5 - 7 + 2 + 0 - 9 + 16 + 10 + 7)/8 = 3\). Then, calculate the squared differences from the mean and get the average of those numbers. These lead to variance of \((2^2 + 10^2 + 1^2 + 3^2 + 12^2 + 13^2 + 7^2 + 4^2) / 8 = 72.5\)
03

Calculate the Standard Deviation

The standard deviation is the square root of the variance. Therefore, take the square root of the variance calculated in the previous step. So the standard deviation equals \(\sqrt{72.5} = 8.52\) (rounded to two decimal places)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Range
The range is a simple way to measure how spread out your data is. It is found by subtracting the smallest value in your dataset from the largest value.
In the dataset given, the largest number is 16, and the smallest is -9. The calculation looks like this:
  • Find the largest number: 16
  • Find the smallest number: -9
  • Subtract them: 16 - (-9) = 25
The range tells us the total spread covered by the data, which in this case, covers 25 units.
Variance
Variance gives us the average of the squared differences between each data point and the mean. It's about understanding how far each number in the set is from the mean.
Here's the breakdown:
  • Calculate the mean: Add all numbers and divide by the count \(\frac{5 - 7 + 2 + 0 - 9 + 16 + 10 + 7}{8} = 3\)

  • Subtract the mean from each number, then square the result:
    • (5 - 3)虏 = 4
    • ((-7) - 3)虏 = 100
    • (2 - 3)虏 = 1

    • (0 - 3)虏 = 9
    • ((-9) - 3)虏 = 144
    • (16 - 3)虏 = 169
    • (10 - 3)虏 = 49
    • (7 - 3)虏 = 16

  • Compute the mean of these squared differences:\(\frac{4 + 100 + 1 + 9 + 144 + 169 + 49 + 16}{8} = 72.5\)
This variance of 72.5 shows us that the data points are quite spread out around the mean.
Standard Deviation
The standard deviation is a measure that shows how much variance or dispersion there is from the average. It is the square root of the variance.

To find the standard deviation:
  • Use the variance from earlier: 72.5
  • Take the square root:\(\sqrt{72.5} \approx 8.52\)
This result of 8.52 tells us that, on average, the numbers in the dataset deviate from the mean by about 8.52 units. This means that if the standard deviation is low, the values tend to be close to the mean. Here, a higher standard deviation suggests significant variation within the dataset.

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Most popular questions from this chapter

The following data give the number of patients who visited a walk-in clinic on each of 24 randomly selected days. \(\begin{array}{lllllllllllr}23 & 37 & 26 & 19 & 33 & 22 & 30 & 42 & 24 & 26 & 64 & 8 \\ 28 & 32 & 37 & 29 & 38 & 24 & 35 & 20 & 34 & 38 & 28 & 16\end{array}\) Prepare a box-and-whisker plot. Comment on the skewness of these data.

The following data give the number of patients who visited a walk-in clinic on each of 20 randomly selected days. \(\begin{array}{llllllllll}23 & 37 & 26 & 19 & 33 & 22 & 30 & 42 & 24 & 26 \\\ 28 & 32 & 37 & 29 & 38 & 24 & 35 & 20 & 34 & 38\end{array}\) a. Calculate the range, variance, and standard deviation for these data. b. Calculate the coefficient of variation.

A large population has a bell-shaped distribution with a mean of 310 and a standard deviation of 37 . Using the empirical rule, find the approximate percentage of the observations that fall in the intervals \(\mu \pm 1 \sigma, \mu \pm 2 \sigma\), and \(\mu \pm 3 \sigma\).

Which of the five measures of center (the mean, the median, the trimmed mean, the weighted mean, and the mode) can be calculated for quantitative data only, and which can be calculated for both quantitative and qualitative data? Illustrate with examples.

The following data give the time (in minutes) that each of 20 students selected from a university waited in line at their bookstore to pay for their textbooks in the beginning of the Fall 2015 semester. \(\begin{array}{rrrrrrrrrr}15 & 8 & 23 & 21 & 5 & 17 & 31 & 22 & 34 & 6 \\ 5 & 10 & 14 & 17 & 16 & 25 & 30 & 3 & 31 & 19\end{array}\) Prepare a box-and-whisker plot. Comment on the skewness of these data.
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