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Chapter 3: Problem 88

Construct two sets of numbers with at least five numbers in each set with the following characteristics: The mean of set \(\mathrm{A}\) is smaller than that of set \(\mathrm{B}\), but the median of set \(\mathrm{B}\) is smaller than that of set \(\mathrm{A}\). Report the mean and the median of both sets of data.

Short Answer

Expert verified
For the given exercise, two feasible sets can be A = {1, 2, 3, 4, 5} and B = {2, 2, 2, 8, 9}. The mean of set A is 3 and the median is also 3. The mean of set B is 4.6 and the median is 2. Therefore, the conditions are satisfied.

Step by step solution

01

Define set A with specific numbers

Consider the first set A as \(A = \{1, 2, 3, 4, 5\}\). When we calculate the mean we get \( \frac{1+2+3+4+5}{5} = 3 \). The median, being the middle value of the set, is also 3.
02

Define set B taking into account both mean and median requirements

Now, for set B, select values that meet the given conditions, i.e., the mean should be greater than that of set A (i.e., > 3), while the median should be less than that of set A (i.e., < 3). A suitable set B under these conditions can be \(B = \{2, 2, 2, 8, 9\}\). In this case, the mean is \( \frac{2+2+2+8+9}{5} = 4.6 \) and the median is 2.
03

Verify if the conditions are met

On comparing the mean and median values for both sets, it's evident that 'mean of set B > mean of set A' and 'median of set B < median of set A'. Hence, the sets A and B are constructed successfully as per the given conditions.

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Key Concepts

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

Descriptive Statistics
Descriptive statistics is a branch of statistics that focuses on summarizing or describing a set of data points. It involves various techniques and measures to provide a summary of the characteristics of the data. These measures can include, but are not limited to, calculations of central tendency, spread, and general distribution of values within a dataset. The goal is to convey information about the sample in a manageable, understandable form, usually for the purpose of making an initial assessment of the data before applying more complex statistical analysis.
Measures of Central Tendency
The measures of central tendency describe the center of a data set or where the bulk of the data values lie. The most common measures include the mean, median, and mode. Each measure provides different insights:
  • Mean: The arithmetic average of all data points.
  • Median: The middle value when the data set is ordered from least to greatest.
  • Mode: The most frequently occurring value in the data set.
These measures help us understand the typical value one might expect from the data set and they each have their own strengths and weaknesses depending on the nature of the data and its distribution.
Constructing Number Sets
When constructing number sets, especially for a specific purpose like comparing mean and median, careful consideration of the values chosen is vital. A set must satisfy the required conditions, such as certain relationships between the mean and median values. Selecting strategic values will allow the sets to fulfill these conditions. For instance, to increase the mean without greatly affecting the median, adding larger numbers at the tail of the set can effectively shift the mean upward while keeping the median about the same position within the ordered set.
Mean Calculation
To calculate the mean of a set of numbers, you sum up all the individual values and then divide by the total number of values in the set. The formula can be expressed as:
\[ \text{Mean} = \frac{\text{Sum of all elements in the set}}{\text{Number of elements in the set}} \].
It's an arithmetic average that can be highly influenced by extreme values or outliers, which may or may not be representative of the dataset as a whole. Therefore, understanding how each value in the number set affects the mean is important for a thorough analysis.
Median Calculation
The median is calculated by first ordering the elements of a number set from smallest to largest and then identifying the middle element. If the set has an odd number of elements, the median is the single middle value. If the set has an even number of elements, the median is the average of the two middle values. It can be expressed as:
\[ \text{Median of odd set} = \text{middle value} \]
\[ \text{Median of even set} = \frac{\text{(n/2)-th value + (n/2 + 1)-th value}}{2} \].
Unlike the mean, the median is less sensitive to extreme values or outliers within a data set, making it a preferred measure of central tendency when dealing with skewed distributions.

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Most popular questions from this chapter

An exam score has a mean of 80 and a standard deviation of 4 . a. Find and interpret in context an exam score that corresponds with a \(z\) -score of 2 b. What exam score corresponds with a \(z\) -score of \(-1.5\) ?

Babies born weighing 2500 grams (about \(5.5\) pounds) or less are called low- birthweight babies, and this condition sometimes indicates health problems for the infant. The mean birth weight for U.S.-bom children is about 3462 grams (about \(7.6\) pounds). The mean birth weight for babies bom one month early is 2622 grams. Suppose both standard deviations are 500 grams. Also assume that the distribution of birth weights is roughly unimodal and symmetric. a. Find the standardized score \((z\) -score), relative to all U.S. births, for a baby with a birth weight of 2500 grams. b. Find the standardized score for a birth weight of 2500 grams for a child born one month early, using 2622 as the mean. c. For which group is a birth weight of 2500 grams more common? Explain what that implies. Unusual \(z\) -scores are far from \(0 .\)

Name two measures of the center of a distribution, and state the conditions under which each is preferred for describing the typical value of a single data set.

In the most recent summer Olympics, do you think the standard deviation of the running times for all men who ran the 100 -meter race would be larger or smaller than the standard deviation of the running times for the men's marathon? Explain.

The tuition costs (in dollars) for a sample of four-year state colleges in California and Texas are shown below. Compare the means and the standard deviations of the data and compare the state tuition costs of the two states in a sentence or two. CA: \(7040,6423,6313,6802,7048,7460\) TX: \(7155,7504,7328,8230,7344,5760\)
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