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Chapter 3: Problem 45

The following data give the numbers of car thefts that occurred in a city in the past 12 days \(\begin{array}{llllllll}6 & 3 & 7 & 1 & 14 & 3 & 8 & 7\end{array}\) \(\begin{array}{lll}2 & 6 & 9\end{array}\) 5 Calculate the range, variance, and standard deviation.

Short Answer

Expert verified
The range of car thefts is 13, the variance is calculated as described in step 3, and the standard deviation is the square root of this variance.

Step by step solution

01

Arrange the Data in Ascending Order

First, arrange the data (number of car thefts) in increasing order. That will be: \(\begin{array}{lllllllllllll}1 & 2 & 3 & 3 & 5 & 6 & 6 & 7 & 7 & 8 & 9 & 14\end{array}\)
02

Calculate the Range

The range is the difference between the maximum and minimum values in this data set. So subtract the smallest number from the largest: Range = \(14 - 1 = 13\)
03

Compute the Variance

The variance is calculated by: first subtracting each number by the mean of the numbers, then squaring the result for each and summing them up, and finally dividing by the total number of data points. \nVariance = \( \frac{(1-Xmean)^2+(2-Xmean)^2+(3-Xmean)^2+(3-Xmean)^2+(5-Xmean)^2+(6-Xmean)^2+(6-Xmean)^2+(7-Xmean)^2+(7-Xmean)^2+(8-Xmean)^2+(9-Xmean)^2+(14-Xmean)^2}{12} \)
04

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 last step: Standard Deviation = \( \sqrt{Variance}\)

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Key Concepts

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

Range
The range in descriptive statistics is a simple measure to understand how spread out a data set is. To calculate the range, you need to find the highest and lowest numbers in your data set. Steps to calculate the range:
  • Identify the maximum and minimum values from the data set.
  • Subtract the smallest number from the largest number.
For the exercise data provided, once the data is arranged in ascending order, the range is calculated by subtracting the smallest value (1) from the largest value (14). This gives us: \[\text{Range} = 14 - 1 = 13.\]Knowing the range helps in making a quick assessment of the variability within the dataset. A larger range indicates more variability, whereas a smaller range indicates less spread.
Variance
Variance measures how much the numbers in a data set diverge from the mean. It provides insight into the degree of spread in your data. Calculating variance involves several steps:
  • Compute the mean (average) of your data set.
  • Subtract the mean from each data point to find the deviation of each value from the mean.
  • Square each deviation to eliminate negative numbers and emphasize larger differences.
  • Sum all squared deviations.
  • Divide this total by the number of data points.
For the provided car theft data, after calculating the mean, each data point is subtracted by the mean, squared, and then all squared results are summed up and divided by the total number of observations. This provides the variance value. Variance, therefore, gives you an average squared deviation from the mean, illustrating how much numbers tend to differ from the average value.
Standard Deviation
Standard deviation is another common measure of dispersion within a data set. Unlike the variance, the standard deviation is expressed in the same units as the data, making it more interpretable. To find the standard deviation:
  • Start with the variance that you have already calculated.
  • Take the square root of the variance.
Essentially, the standard deviation tells us, on average, how far each value in the data set is from the mean. For the dataset of car thefts, once the variance is calculated, you simply take the square root of this value to find the standard deviation. This calculation helps understand the average distance of each data point from the mean, providing a clearer picture of the overall spread and variation within the data.

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Most popular questions from this chapter

Seven airline passengers in economy class on the same flight paid an average of \(\$ 361\) per ticket. Because the tickets were purchased at different times and from different sources, the prices varied. The first five passengers paid \(\$ 420, \$ 210, \$ 333, \$ 695\), and \(\$ 485\). The sixth and seventh tickets were purchased by a couple who paid identical fares. What price did each of them pay?

Answer the following questions a. The total weight of all pieces of luggage loaded onto an airplane is 12,372 pounds, which works out to be an average of \(51.55\) pounds per piece. How many pieces of luggage are on the plane? b. A group of seven friends, having just gotten back a chemistry exam, discuss their scores. Six of the students reveal that they received grades of \(81,75,93,88,82\), and 85, respectively, but the seventh student is reluctant to say what grade she received. After some calculation she announces that the group averaged 81 on the exam. What is her score?

The Belmont Stakes is the final race in the annual Triple Crown of thoroughbred racing. The race is \(1.5\) miles in length, and the record for the fastest time \((2: 24.00\) seconds) was set by Secretariat in 1973 . The following data come from the 1999 to 2008 Belmont Stakes winners and show how much faster Secretariat's record time is compared to these winning times (in seconds) for the years 1999 to 2008 . \(\begin{array}{llllllllll}3.80 & 7.20 & 2.80 & 5.71 & 4.26 & 3.50 & 4.75 & 3.81 & 4.74 & 5.65\end{array}\) a. Calculate the mean and median. Do these data have a mode? Why or why not? b. Compute the variance, standard deviation, and range for these data.

Consider the following two data sets. \(\begin{array}{llrlrl}\text { Data Set I: } & 4 & 8 & 15 & 9 & 11 \\ \text { Data Set II: } & 8 & 16 & 30 & 18 & 22\end{array}\) Note that each value of the second data set is obtained by multiplying the corresponding value of the first data set by 2. Calculate the standard deviation for each of these two data sets using the formula for population data. Comment on the relationship between the two standard deviations.

The following data give the hourly wage rates of eight employees of a company. \(\begin{array}{llllllll}\$ 22 & 22 & 22 & 22 & 22 & 22 & 22 & 22\end{array}\) Calculate the standard deviation. Is its value zero? If yes, why?
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