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Chapter 4: Problem 17

The following data are cost (in cents) per ounce for nine different brands of sliced Swiss cheese (www .consumerreports.org): \(\begin{array}{llllllll}29 & 62 & 37 & 41 & 70 & 82 & 47 & 52\end{array}\) a. Compute the variance and standard deviation for this data set. \(s^{2}=279.111 ; s=16.707\) b. If a very expensive cheese with a cost per slice of 150 cents was added to the data set, how would the values of the mean and standard deviation change?

Short Answer

Expert verified
The original Variance of the data was 279.111 and the Standard Deviation was 16.707. After adding a new data point of 150, the Variance changed to 1395.492, and the Standard Deviation increased to 37.377.

Step by step solution

01

Computation of Mean

First, one needs to calculate the mean of the set. The \(\text{Mean}\) or \(\text{average}\) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set. For the given set of data, the \(\text{Mean}\) can be calculated as \( \frac{29+62+37+41+70+82+47+52}{8} \). This gives the mean as 52.5.
02

Calculation of Variance

The \(\text{Variance}\) (\(s^{2}\)) is the average of the squared differences from the Mean, calculated as \( \frac{{(29-52.5)^{2}+(62-52.5)^{2}+(37-52.5)^{2}+(41-52.5)^{2}+(70-52.5)^{2}+(82-52.5)^{2}+(47-52.5)^{2}+(52-52.5)^{2}}}{8-1} \). This gives the Variance as 279.111.
03

Calculation of Standard Deviation

The \(\text{Standard Deviation}\) (\(s\)) is simply the square root of the variance, that is \( \sqrt{279.111} = 16.707 \).
04

Adding a Value

Then we introduce a new value (150) to the data set. This will affect the values of the mean and standard deviation. The new mean can be calculated as \( \frac{29+62+37+41+70+82+47+52+150}{9} = 63.222 \). Then we can calculate the new standard deviation similarly as before by using the new mean in variance formula.
05

Calculation of Variance with New Mean

In this step, the new variance is calculated with new mean and data set. The new variance can be calculated as \( \frac{{(29-63.222)^{2}+(62-63.222)^{2}+(37-63.222)^{2}+(41-63.222)^{2}+(70-63.222)^{2}+(82-63.222)^{2}+(47-63.222)^{2}+(52-63.222)^{2}+(150-63.222)^{2}}}{9-1} \). The new variance is 1395.492.
06

Calculation of New Standard Deviation

The new standard deviation is the square root of the new variance, looking as \( \sqrt{1395.492} = 37.377 \).

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Key Concepts

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

Mean
Calculating the "mean" of a data set is a fundamental concept in statistics. It's the "average" of the numbers in your data set. To calculate it, you simply add up all the numbers in your data set and divide by the number of entries. In the given example of Swiss cheese costs:
  • Our numbers are: 29, 62, 37, 41, 70, 82, 47, and 52.
  • Add them up to get a total of 420.
  • Then, divide the total by the number of data points, which is 8.
  • This gives a mean of 52.5.
So, the mean provides a central value for the data set, giving us an idea of where most data points tend to cluster.
Variance
Variance is a measure that tells us how much our data points differ from the mean. In simpler terms, it quantifies the spread of the data set. To calculate variance:
  • First, find the differences between each data point and the mean, and then square these differences. Squaring ensures all values are positive, emphasizing larger deviations.
  • Next, average these squared differences. However, instead of dividing by the number of data points (as done in calculating mean), we divide by the number of data points minus one. This accounts for sample variance, which provides a slightly more reliable estimate for larger populations.
In our example:
  • The variance is 279.111, which indicates a moderate spread of data points around the mean of 52.5.
Variance gives us insight into the data's variability, which is crucial for understanding consistency within the dataset.
Standard Deviation
The "standard deviation" is another measure of how spread out the numbers in a data set are. It's more intuitive than variance because it is in the same unit as our original data. To calculate the standard deviation:
  • Simply take the square root of the variance.
For our original cheese cost example:
  • With a variance of 279.111, the standard deviation equals the square root of 279.111, which results in 16.707.
A small standard deviation means that most of the numbers are close to the mean. Conversely, a larger standard deviation indicates that the numbers are more spread out from the mean. Here, a standard deviation of 16.707 shows a considerable variability in the cheese cost data set.
Data Set
In statistics, a "data set" consists of collections of data points or values. Understanding a data set's characteristics helps make sense of its analysis and the implications of adding or removing data points. Our data set originally includes: 29, 62, 37, 41, 70, 82, 47, and 52.
  • When considering how adding data can affect key statistics like mean and standard deviation, it's crucial to update calculations each time the data set changes. For example, introducing a new brand costing 150 cents drastically increased the mean (from 52.5 to 63.222) and the standard deviation (from 16.707 to 37.377).
This demonstrates that any changes to the data set will alter the summary statistics, potentially affecting conclusions drawn from the data. Recognizing the impact of these changes allows for a more comprehensive analysis and interpretation.

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