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Chapter 2: Problem 74

Consider the sample \(2,4,7,8,9 .\) Find the following: a. \(\operatorname{mean}, \bar{x}\) b. \(\operatorname{median}, \widetilde{x}\) c. mode d. midrange

Short Answer

Expert verified
The mean \(\bar{x}\) is 6, the median \(\widetilde{x}\) is 7, there is no mode and the midrange is 5.5.

Step by step solution

01

Calculation of mean

The mean \(\bar{x}\) is the average of all numbers and is calculated by adding all numbers and dividing by the count of numbers. \(\bar{x}=\frac{2+4+7+8+9}{5}=6\)
02

Calculation of median

The median \(\widetilde{x}\) is the middle number in an ordered list. First, rearrange the numbers in ascending order: \(2,4,7,8,9\). Then, since there are 5 numbers, the median is the third number. \(\widetilde{x}=7\)
03

Find The Mode

The mode is the number that appears most frequently in a data set. A data set may have one mode, more than one mode, or no mode at all. Since all numbers in our data set are unique, this set has no mode.
04

Calculation of Midrange

The midrange is calculated as the average of the maximum and minimum values of a data set. \(\text{Midrange}=\frac{\text{Max value + Min value}}{2}\) \(\text{Midrange}=\frac{9+2}{2}=5.5\)

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Key Concepts

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

Mean Calculation
The mean, often referred to as the average, represents the sum of all data points in a set divided by the number of points. To calculate the mean of a sample, you should follow a simple step-by-step process: Add all the numbers in the list together, then divide by how many numbers there are in total.

For our sample \(2, 4, 7, 8, 9\), the calculation looks like this:
  • First, add 2 + 4 + 7 + 8 + 9 = 30.
  • Then, divide 30 by the number of items, which is 5.
  • This gives us the mean: \(\bar{x} = 6\).
Understanding the mean can help you get a sense of the overall "average" value within a dataset.
Median Calculation
The median provides the central value in a data set when it's arranged from lowest to highest. It helps give a picture of the data's distribution with one single number. When you have an odd number of data points, the median is simply the middle one.

Our sample is \(2, 4, 7, 8, 9\). The calculation of the median involves these steps:
  • List all numbers in ascending order: 2, 4, 7, 8, 9.
  • The number in the middle, which is the third number here, is 7.
  • Thus, the median \(\widetilde{x} = 7\).
This approach shows you the center position in your data, unaffected by extreme values.
Mode Determination
The mode reflects the number that occurs the most frequently in a data set. It's quite straightforward but does require you to count the occurrences of each number.

In a scenario where each number is unique, as seen in the set \(2, 4, 7, 8, 9\), no number repeats. This means that our data set doesn't have a mode.

However, it's possible to have:
  • One mode when a single number appears more often than others.
  • Multiple modes if several numbers have the same, maximum frequency.
  • No mode if no number repeats.
Recognizing the mode, or knowing that there isn't one, offers insights into data uniformity.
Midrange Calculation
The midrange is a measure of central tendency that averages the smallest and largest numbers in your dataset. It provides a quick sense of the data鈥檚 spread.

Here鈥檚 how you can calculate the midrange for our example \(2, 4, 7, 8, 9\):
  • Identify the minimum value, which is 2, and the maximum value, which is 9.
  • Add these two values: 9 + 2 = 11.
  • Divide the result by 2 to find the midrange: \(\frac{11}{2} = 5.5\).
The midrange can give you a balanced view of your dataset's range, taking both extremes into account.

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Most popular questions from this chapter

All measures of variation are nonnegative in value for all sets of data. a. What does it mean for a value to be "nonnegative"? b. Describe the conditions necessary for a measure of variation to have the value zero. c. Describe the conditions necessary for a measure of variation to have a positive value.

Why is it that the \(z\) -score for a value that belongs to a normal distribution usually lies between -3 and \(+3 ?\)

Adding (or subtracting) the same number from each value in a set of data does not affect the measures of variability for that set of data. a. Find the variance of this set of annual heatingdegree-day data: 6017,6173,6275,6350,6001,6300. b. Find the variance of this set of data (obtained by subtracting 6000 from each value in part a): 17,173 275,350,1,300.

The opening-round scores for the Ladies' Professional Golf Association tournament at Locust Hill Country Club were posted as follows: $$\begin{array}{llllllllllllll} \hline 69 & 73 & 72 & 74 & 77 & 80 & 75 & 74 & 72 & 83 & 68 & 73 & 75 & 78 \\ 76 & 74 & 73 & 68 & 71 & 72 & 75 & 79 & 74 & 75 & 74 & 74 & 68 & 79 \\ 75 & 76 & 75 & 77 & 74 & 74 & 75 & 75 & 72 & 73 & 73 & 72 & 72 & 71 \\ 71 & 70 & 82 & 77 & 76 & 73 & 72 & 72 & 72 & 75 & 75 & 74 & 74 & 74 \\ 76 & 76 & 74 & 73 & 74 & 73 & 72 & 72 & 74 & 71 & 72 & 73 & 72 & 72 \\ 74 & 74 & 67 & 69 & 71 & 70 & 72 & 74 & 76 & 75 & 75 & 74 & 73 & 74 \\ 74 & 78 & 77 & 81 & 73 & 73 & 74 & 68 & 71 & 74 & 78 & 70 & 68 & 71 \\ 72 & 72 & 75 & 74 & 76 & 77 & 74 & 74 & 73 & 73 & 70 & 68 & 69 & 71 \\ 77 & 78 & 68 & 72 & 73 & 78 & 77 & 79 & 79 & 77 & 75 & 75 & 74 & 73 \\ 73 & 72 & 71 & 68 & 70 & 71 & 78 & 78 & 76 & 74 & 75 & 72 & 72 & 72 \\ 75 & 74 & 76 & 77 & 78 & 78 & & & & & & & \\ \hline \end{array}$$ a. Form an ungrouped frequency distribution of these scores. b. Draw a histogram of the first-round golf scores. Use the frequency distribution from part a.

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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