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Chapter 2: Problem 183

For each of the data sets in parts \(\mathbf{a}-\mathbf{c},\) compute \(\bar{x}, s^{2},\) and \(s\) If appropriate, specify the units in which your answers are expressed. a. 4,6,6,5,6,7 b. \(-\$ 1, \$ 4,-\$ 3, \$ 0,-\$ 3,-\$ 6\) c. \(3 / 5 \%, 4 / 5 \%, 2 / 5 \%, 1 / 5 \%, 1 / 16 \%\) d. Calculate the range of each data set in parts \(\mathbf{a}-\mathbf{c}\).

Short Answer

Expert verified
a: \(\bar{x}=5.67\), \(s^2=1.07\), \(s=1.03\); b: \(\bar{x}=-1.5\), \(s^2=9.9\), \(s=3.15\); c: \(\bar{x}=0.4125\%\), \(s^2=0.08469\), \(s=0.291\%\); Ranges: a: 3, b: \$10, c: 0.7375%

Step by step solution

01

Compute the Mean for Set a

For data set a: 4, 6, 6, 5, 6, 7, compute the mean (\(\bar{x}\)) using the formula: \[ \bar{x} = \frac{\sum{x_i}}{n} \], where \(n\) is the number of data points. So, \(\bar{x} = \frac{4 + 6 + 6 + 5 + 6 + 7}{6} = 5.67\).
02

Compute the Variance for Set a

The variance \(s^2\) is calculated using the formula: \[ s^2 = \frac{\sum{(x_i - \bar{x})^2}}{n-1} \]. Substituting the values: \[ s^2 = \frac{(4-5.67)^2 + (6-5.67)^2 + (6-5.67)^2 + (5-5.67)^2 + (6-5.67)^2 + (7-5.67)^2}{5} = 1.07 \].
03

Compute the Standard Deviation for Set a

The standard deviation \(s\) is the square root of the variance: \[ s = \sqrt{s^2} = \sqrt{1.07} \approx 1.03 \].
04

Compute Mean, Variance, and Standard Deviation for Set b

Compute the mean: \(\bar{x} = \frac{-1 + 4 - 3 + 0 - 3 - 6}{6} = -\frac{9}{6} = -1.5\). The variance \(s^2\) is \[ s^2 = \frac{(-1 + 1.5)^2 + (4 + 1.5)^2 + (-3 + 1.5)^2 + (0 + 1.5)^2 + (-3 + 1.5)^2 + (-6 + 1.5)^2}{5} = 9.9\]. The standard deviation \(s = \sqrt{9.9} \approx 3.15\). Units are dollars.
05

Compute Mean, Variance, and Standard Deviation for Set c

Convert percentages to decimals: \(\frac{3}{5}\%, \frac{4}{5}\%, \frac{2}{5}\%, \frac{1}{5}\%, \frac{1}{16}\%\) becomes \(0.6\%, 0.8\%, 0.4\%, 0.2\%, 0.0625\%\). Compute the mean: \(\bar{x} = \frac{0.6 + 0.8 + 0.4 + 0.2 + 0.0625}{5} \approx 0.4125\%\). The variance \(s^2\) is \(s^2 = \frac{(0.6-0.4125)^2+(0.8-0.4125)^2+(0.4-0.4125)^2+(0.2-0.4125)^2+(0.0625-0.4125)^2}{4} \approx 0.08469\). The standard deviation \(s = \sqrt{0.08469} \approx 0.291\%\).
06

Calculate the Range for Each Data Set

The range is the difference between the maximum and minimum values.- For set a: \(7 - 4 = 3\).- For set b: \(\\(4 - (-\\)6) = \$10\).- For set c: \(0.8\% - 0.0625\% = 0.7375\%\).

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Key Concepts

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

Mean Calculation
Calculating the mean, also known as the average, is a fundamental step in descriptive statistics. The mean provides a central value that represents a data set. To calculate the mean of a data set, sum all the values and then divide by the number of values. This formula looks like this: \[ \bar{x} = \frac{\sum{x_i}}{n} \] where \( \sum{x_i} \) is the sum of all data points and \( n \) is the number of data points. For example, for data set \( a \) (4, 6, 6, 5, 6, 7), the sum is 34, and since there are 6 numbers, the mean is \( \frac{34}{6} = 5.67 \). This gives us an idea of the average value of this data set.
Variance and Standard Deviation
Variance and standard deviation are measures of spread, showing how much the data values deviate from the mean.
  • Variance is calculated by taking the average of the squared deviations from the mean. This is done using the formula: \[ s^2 = \frac{\sum{(x_i - \bar{x})^2}}{n-1} \] where \( x_i \) are the data points. For data set \( a \), this evaluates to \( 1.07 \), showing how much the numbers spread out around the mean.
  • Standard deviation is simply the square root of the variance: \[ s = \sqrt{s^2} \]. For set \( a \), this is approximately 1.03. The standard deviation provides insight into the average distance of the data points from the mean, thus giving a sense of the data's variability.
These concepts help to understand not just the central tendency, but also the distribution of the data points.
Range Calculation
The range is one of the simplest measures of dispersion and indicates the difference between the largest and smallest values in a data set. It's a quick way to get a sense of how spread out the values are.
  • For data set \( a \) (4, 6, 6, 5, 6, 7), the range is calculated as \( 7 - 4 = 3 \).
  • For data set \( b \) (-1, 4, -3, 0, -3, -6), it is \( 4 - (-6) = 10 \), giving us an \$ range again showing the extent of variation in dollar terms.
  • For the percentage data set \( c \) (0.6%, 0.8%, 0.4%, 0.2%, 0.0625%), the range is \( 0.8\% - 0.0625\% = 0.7375\% \).
While the range is easy to calculate, it doesn't provide much detail about the data spread beyond the extreme values.

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Most popular questions from this chapter

Graph the relative frequency histogram for the 500 measurements summarized in the accompanying relative frequency table.$$ \begin{array}{cc} \text { Class Interval } & \text { Relative Frequency } \\ \hline .5-2.5 & .10 \\ 2.5-4.5 & .15 \\ 4.5-6.5 & .25 \\ 6.5-8.5 & .20 \\ 8.5-10.5 & .05 \\ 10.5-12.5 & .10 \\ 12.5-14.5 & .10 \\ 14.5-16.5 & .05 \end{array} $$

Refer to the American Journal of Physical Anthropology (Vol. 142,2010 ) study of the characteristics of cheek teeth (e.g., molars) in an extinct primate species, Exercise \(2.65(\mathrm{p} .90) .\) The data on dentary depth of molars (in millimeters) for 18 cheek teeth extracted from skulls are reproduced in the table.$$ \begin{array}{lcc} \hline \text { Data on Dentary Depth (mm) of Molars } & \\ \hline 18.12 & 15.76 & 13.25 \\ 19.48 & 17.00 & 16.12 \\ 19.36 & 13.96 & 18.13 \\ 15.94 & 16.55 & 14.02 \\ 15.83 & 15.70 & 14.04 \\ 19.70 & 17.83 & 16.20 \end{array} $$ a. Find the range of the data set. If the largest depth measurement in the sample were doubled, how would the range change? Would it increase or decrease? b. Find the variance of the data set. If the largest depth measurement in the sample were doubled, how would the variance change? Would it increase or decrease? c. Find the standard deviation of the data set. If the largest depth measurement in the sample were doubled, how would the standard deviation change? Would it increase or decrease?

Why do we square the deviation from the mean before adding them to compute the variance?

Discuss conditions under which the median is preferred to the mean as a measure of central tendency.

Which of the three measures of central tendency may not be unique for a given data set?
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