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Chapter 2: Problem 175

Consider samples \(A\) and \(B\). Notice that the two samples are the same except that the 8 in A has been replaced by a 9 in B. $$\begin{array}{lllllll} \hline \mathbf{A} & 2 & 4 & 5 & 5 & 7 & 8 \\ \mathbf{B} & 2 & 4 & 5 & 5 & 7 & 9 \\ \hline \end{array}$$ What effect does changing the 8 to a 9 have on each of the following statistics? a. Mean b. Median c. Mode d. Midrange e. Range f. Variance g. Std. dev.

Short Answer

Expert verified
Changing '8' to '9' increases the mean from 5.16 to 5.33, the midrange from 5 to 5.5, the range from 6 to 7, the variance from 4.97 to 5.33, and the standard deviation from 2.23 to 2.31. However, the median and mode remain the same.

Step by step solution

01

Calculate the Original Values

Start by calculating the original values for A first: a. Mean: \((2+4+5+5+7+8)/6=5.16\); b. Median: The median value is the average of the middle two numbers, i.e., \((5+5)/2=5\); c. Mode: The mode is the number that appears most frequently, which is \(5\); d. Midrange: The midrange is the average of the smallest and largest numbers, \((2+8)/2=5\); e. Range: The range is the difference between the smallest and largest numbers, \(8-2=6\); f. Variance: The variance is the average of the squared differences from the mean, which results in \(4.97\); g. Standard deviation: The standard deviation is the square root of the variance, which equals to \(2.23\).
02

Calculate the New Values

Now calculate the values for B after replacing '8' with '9': a. Mean: \((2+4+5+5+7+9)/6=5.33\); b. Median: The median doesn't change as it still averages the middle two numbers, i.e., \((5+5)/2=5\); c. Mode: The mode remains at \(5\); d. Midrange: This is now \((2+9)/2=5.5\); e. Range: The range is now \(9-2=7\); f. Variance: The variance changes to \(5.33\); g. Standard deviation: This changes to \(2.31\).
03

Determine the Impact of the Change

Comparing these sets of values, we can see that changing the '8' to a '9' resulted in changes to the mean, midrange, range, variance, and standard deviation, but the median and mode were unaffected.

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Key Concepts

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

Understanding the Mean
The mean is a fundamental concept in statistics used to find the average value of a dataset. It is calculated by adding up all the numbers and then dividing by the total count of the numbers. For sample A, the mean is calculated as:
  • Step 1: Add all numbers: \(2 + 4 + 5 + 5 + 7 + 8 = 31\)
  • Step 2: Divide by the total number of values: \( \frac{31}{6} = 5.16\)
When the 8 in Sample A is replaced by 9 in Sample B, the sum becomes 32. Thus, the new mean is:
  • Step 1: New sum: \(2 + 4 + 5 + 5 + 7 + 9 = 32\)
  • Step 2: Divide by the total number of values: \( \frac{32}{6} = 5.33\)
This slight increase from 5.16 to 5.33 shows how sensitive the mean is to changes in the data values.
Exploring the Median
The median is a simple measure that represents the "middle" value in an ordered dataset, providing a measure of central tendency that is not significantly affected by outliers or skewed data. In our samples:
  • To find the median, first arrange the numbers in order.
  • For both Sample A and Sample B, the ordered numbers are: 2, 4, 5, 5, 7, and 8 (or 9 for B).
  • With an even number of observations, the median is the average of the two middle numbers.
  • Average of middle numbers for both samples: \( \frac{5 + 5}{2} = 5\)
The median remains unchanged at 5 even when one number in the sample is replaced, demonstrating its stability and resilience compared to other measures such as the mean.
Deciphering the Mode
The mode is the simplest measure of central tendency that identifies the most frequently occurring value in a dataset. In both Sample A and Sample B:
  • The number 5 appears more frequently than any other numbers.
  • This makes 5 the mode for both samples \(A\) and \(B\).
  • The replacement of 8 with 9 does not affect the mode since neither 8 nor 9 appear more frequently than 5.
This illustrates how the mode can provide useful insights about the distribution even when data changes do not affect it significantly.
Diving into Variance
Variance is a measure of how much the values in a dataset spread out from their mean, offering insight into the variability of the data. It is calculated by averaging the squared differences between each data point and the mean.
  • First, calculate each distance from the mean and square it.
  • For Sample A (mean = 5.16), the square of differences lead to a variance of approximately 4.97.
  • For Sample B (mean = 5.33), the new square of differences results in a slightly higher variance of 5.33.
This increase identifies how a change in even a single data point can alter the overall spread in the data, making variance a vital statistic for understanding dataset variability.

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Most popular questions from this chapter

The Office of Coal, Nuclear, Electric and Alternate Fuels reported the following data as the costs (in cents) of the average revenue per kilowatt- hour for sectors in Arkansas: $$\begin{array}{lllllllll} \hline 6.61 & 7.61 & 6.99 & 7.48 & 5.10 & 7.56 & 6.65 & 5.93 & 7.92 \\ 5.52 & 7.47 & 6.79 & 8.27 & 7.50 & 7.44 & 6.36 & 5.20 & 5.48 \\ 7.69 & 8.74 & 5.75 & 6.94 & 7.70 & 6.67 & 4.59 & 5.96 & 7.26 \\ 5.38 & 8.88 & 7.49 & 6.89 & 7.25 & 6.89 & 6.41 & 5.86 & 8.04 \\ \hline \end{array}$$ a. Prepare a grouped frequency distribution for the average revenue per kilowatt-hour using class boundaries 4,5,6,7,8,9 b. Find the class width. c. List the class midpoints. d. Construct a relative frequency histogram of these data.

Demonstrates the effect that the number of classes or bins has on the shape of a histogram. a. What shape distribution does using one class or bin produce? b. What shape distribution does using two classes or bins produce? c. What shape distribution does using 10 or 20 bins produce?

For a normal (or bell-shaped) distribution, find the percentile rank that corresponds to: a. \(z=2\) b. \(z=-1\) c. Sketch the normal curve, showing the relationship between the \(z\) -score and the percentiles for parts a

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The number of points scored during each game by a high school basketball team last season was as follows: 56,54,61,71,46,61,55,68,60,66,54,61 \(52,36,64,51 .\) Construct a dotplot of these data.
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