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Chapter 3: Problem 16

In this problem, we explore the effect on the mean, median, and mode of adding the same number to each data value. Consider the data set 2,2,3,6,10. (a) Compute the mode, median, and mean. (b) Add 5 to each of the data values. Compute the mode, median, and mean. (c) Compare the results of parts (a) and (b). In general, how do you think the mode, median, and mean are affected when the same constant is added to each data value in a set?

Short Answer

Expert verified
Adding a constant to each data value increases the mode, median, and mean by that constant.

Step by step solution

01

Calculate the Mode, Median, and Mean of Original Data Set

Given the data set: 2, 2, 3, 6, 10.**Mode**: The mode is the number that appears most frequently. In this case, the mode is 2, as it appears twice.**Median**: Arrange the numbers in ascending order (which they already are in). The median is the middle number. For this data set, the median is 3 (the third number).**Mean**: The mean is calculated by summing all the numbers and dividing by the count. \[\text{Mean} = \frac{2 + 2 + 3 + 6 + 10}{5} = \frac{23}{5} = 4.6\]
02

Add 5 to Each Data Value and Calculate the New Mode, Median, and Mean

Add 5 to each of the data values: 7, 7, 8, 11, 15.**Mode**: The mode is 7, as it appears twice.**Median**: The dataset is again in ascending order. The median is still the middle number, which is 8 (third number).**Mean**: Recalculate the mean after adding 5 to each data value. \[\text{Mean} = \frac{7 + 7 + 8 + 11 + 15}{5} = \frac{48}{5} = 9.6\]
03

Compare and Analyze the Results

In the original data set, we found the mode to be 2, the median to be 3, and the mean to be 4.6. After adding 5, the mode is 7, the median is 8, and the mean is 9.6. **Comparison**: - The mode increased from 2 to 7; the median increased from 3 to 8; the mean increased from 4.6 to 9.6. In general, when the same constant is added to each data value, the mode, median, and mean all increase by that constant.

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Key Concepts

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

Understanding Mode
The mode of a data set is the value that appears most frequently. It gives us a sense of the most common or popular item within a dataset. In the original dataset: 2, 2, 3, 6, 10, the mode is 2, as it appears more frequently than any other number. After adding a constant value of 5 to each data point, the revised data becomes 7, 7, 8, 11, 15, and the mode shifts to 7.
This demonstrates how the mode can shift based on changes to the dataset. It's important to note that while adding a constant to each data point affects the mode, the mode will only change if the distribution of numbers changes as a result.
In summary, when you add a constant to every item in your dataset, the mode will also increase by that same constant. However, the mode can remain unchanged if there's no new frequent value created by this addition.
Exploring Median
The median is the central value of an ordered set of numbers, offering a measure of the data's center that is less affected by outliers than the mean. For the original set 2, 2, 3, 6, 10, the median is 3, as it is the middle value.
When a constant is added to each data point resulting in a new set of 7, 7, 8, 11, 15, the median shifts to 8. Thus, adding 5 to each value has shifted the entire set upwards by 5, including the median.
  • This property of shifting by a constant is crucial when dealing with data transformations.
  • The median, unlike the mean, provides a reliable central tendency measure when the data has outliers or skewed distribution.
This ensures that when every number in a dataset is changed by the same constant, the relative position of data points is maintained.
Calculating Mean
The mean, or average, is calculated by summing all the values in a dataset and dividing by the number of values. It's a common measure of central tendency, showing the typical value within a dataset. For our initial set, we find the mean by calculating: \[\text{Mean} = \frac{2+2+3+6+10}{5} = 4.6\]When 5 is added to each element, the dataset becomes 7, 7, 8, 11, 15. The new mean is calculated as:\[\text{Mean} = \frac{7+7+8+11+15}{5} = 9.6\]Every number in the dataset has increased by 5, causing the mean to increase by the same amount.
This predictable shift confirms that adding a constant to each data value results in the mean increasing by that constant. The mean is useful for datasets with symmetrically distributed data, but it can be skewed by outliers, unlike the median.

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Most popular questions from this chapter

Consider population data with \(\mu=20\) and \(\sigma=2\). (a) Compute the coefficient of variation. (b) Compute an \(88.9 \%\) Chebyshev interval around the population mean.

The Hill of Tara in Ireland is a place of great archaeological importance. This region has been occupied by people for more than 4000 years. Geomagnetic surveys detect subsurface anomalies in the earth's magnetic field. These surveys have led to many significant archaeological discoveries. After collecting data, the next step is to begin a statistical study. The following data measure magnetic susceptibility (centimeter-gram-second \(\times 10^{-6}\) ) on two of the main grids of the Hill of Tara (Reference: Tara: An Archaeological Survey by Conor Newman, Royal Irish Academy, Dublin). Grid \(\mathbf{E}: x\) variable $$\begin{array}{ccccccc} 13.20 & 5.60 & 19.80 & 15.05 & 21.40 & 17.25 & 27.45 \\ 16.95 & 23.90 & 32.40 & 40.75 & 5.10 & 17.75 & 28.35 \end{array}$$ Grid H: \(y\) variable $$\begin{array}{lllllll} 11.85 & 15.25 & 21.30 & 17.30 & 27.50 & 10.35 & 14.90 \\ 48.70 & 25.40 & 25.95 & 57.60 & 34.35 & 38.80 & 41.00 \\ 31.25 & & & & & \end{array}$$ (a) Compute \(\Sigma x, \Sigma x^{2}, \Sigma y,\) and \(\Sigma y^{2}\). (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for \(x\) and for \(y\). (c) Compute a \(75 \%\) Chebyshev interval around the mean for \(x\) values and also for \(y\) values. Use the intervals to compare the magnetic susceptibility on the two grids. Higher numbers indicate higher magnetic susceptibility. However, extreme values, high or low, could mean an anomaly and possible archaeological treasure. (d) Compute the sample coefficient of variation for each grid. Use the \(C V\) s to compare the two grids. If \(s\) represents variability in the signal (magnetic susceptibility) and \(\bar{x}\) represents the expected level of the signal, then \(s / \bar{x}\) can be thought of as a measure of the variability per unit of expected signal. Remember, a considerable variability in the signal (above or below average) might indicate buried artifacts. Why, in this case, would a large \(C V\) be better, or at least more exciting? Explain.

Grades: Weighted Average In your biology class, your final grade is based on several things: a lab score, scores on two major tests, and your score on the final exam. There are 100 points available for each score. However, the lab score is worth \(25 \%\) of your total grade, each major test is worth \(22.5 \%\) and the final exam is worth \(30 \%\). Compute the weighted average for the following scores: 92 on the lab, 81 on the first major test, 93 on the second major test, and 85 on the final exam.

In order to find the median of a data set, what do we do first with the data?

Police are tested for their ability to correctly recognize and identify a suspect based on a witness or victim's verbal description of the suspect. Scores on the identification test range from 0 to 100 (perfect score). Three cities in Massachusetts are under study. The mean score for all police in the three cities is requested. However, funding will only permit \(m=150\) police to be tested. There is no preliminary study to estimate sample standard deviation of scores in each city. City A has \(N_{1}=183\) police, City B has \(N_{2}=371\) police, and City \(\mathrm{C}\) has \(N_{3}=255\) police. In most cases \(n_{i}\) is not a whole number so we round to the nearest whole number. Remember our total sample size is \(m=n_{1}+n_{2}+n_{3}\) (a) Use the method of proportional sampling to compute \(n_{1}, n_{2},\) and \(n_{3},\) the size of the random sample to be taken from each of the three cities. Round each \(n_{i}\) to the nearest whole number and make sure \(m=n_{1}+n_{2}+n_{3}\) (b) Suppose you actually conducted the specified number of tests in each city and obtained the following result: \(\overline{x_{1}}=96\) is the mean test score from city \(\mathrm{A}, \overline{x_{2}}=85\) is the mean test score from City \(\mathrm{B},\) and \(\overline{x_{3}}=88\) is the mean test score from City C. Use the weighted average \(\mu \approx \frac{n_{1}}{m} \overline{x_{1}}+\frac{n_{2}}{m} \overline{x_{2}}+\frac{n_{3}}{m} \overline{x_{3}}\) to get your best estimate for the population mean test score of all police in all three cities. Estimates for \(\mu\) are usually better if we have a preliminary study with reasonably accurate estimates for the sample standard deviation \(s_{i}\) of each stratum (see Problems 28 and 29 ). However, in the case where there is no preliminary study we can use proportional sampling and still obtain good results.
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