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The table in the next column shows the number of times \(20-24\) -year-old U.S. residents have been married, based on a Bureau of the Census report from 2004 . The frequencies are actually thousands of people. For instance, 8,418,000 men never married, but this does not affect calculations about the mean or median. $$\begin{array}{crr} \hline {\text { Number of Times Married, for Subjects of Age 20-24 }} \\ \hline & {\text { Frequency }} \\ \hline \text { Number Times Married } & \text { Women } & {\text { Men }} \\ \hline 0 & 7350 & 8418 \\ 1 & 2587 & 1594 \\ 2 & 80 & 10 \\\\\text { Total } & \mathbf{1 0 , 0 1 7} & \mathbf{1 0 , 0 2 2} \\\ \hline\end{array}$$ a. Find the median and mean for each gender. b. On average, have women or men been married more often? Which statistic do you prefer to answer this question? (The mean, as opposed to the median, uses the numerical values of all the observations, not just the ordering. For discrete data with only a few values such as the number of times married, it can be more informative.)

Step by step solution

01

Identify the Total Number of Subjects

For both women and men, determine the total number of subjects by summing up the frequencies provided in the table. For women, the total is \(7350 + 2587 + 80 = 10017\). For men, the total is \(8418 + 1594 + 10 = 10022\).

02

Find the Mean Number of Times Married

The mean is calculated by \( \text{Mean} = \frac{\Sigma ( \text{Number of Marriages} \times \text{Frequency} )}{\text{Total Subjects}} \).For women:\[ \text{Mean}_{\text{Women}} = \frac{0 \times 7350 + 1 \times 2587 + 2 \times 80}{10017} = \frac{2747}{10017} \approx 0.274 \].For men:\[ \text{Mean}_{\text{Men}} = \frac{0 \times 8418 + 1 \times 1594 + 2 \times 10}{10022} = \frac{1614}{10022} \approx 0.161 \].

03

Find the Median Number of Times Married

The median is the middle value when the numbers are ordered. For both women and men: For women, find the cumulative frequencies: - Never married: 7350 - 1 time: 7350 + 2587 = 9937 - Thus, the median falls in the category "1 time married." For men, find the cumulative frequencies: - Never married: 8418 - 1 time: 8418 + 1594 = 10012 - Thus, the median falls in the category "0 times married."

04

Determine Which Gender Has Been Married More Often on Average

From the calculated means, compare the average number of times married between women and men. - Mean for women is approximately 0.274. - Mean for men is approximately 0.161. Thus, on average, women have been married more often than men. We use the mean because it considers all data points.

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Key Concepts

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

Mean Calculation

A "mean" is what many think of as the "average." It provides a central value by considering all the data points. This can be especially telling when looking at patterns or trends. To calculate the mean number of times someone in our dataset has been married, we multiply each number of marriages by how often it occurs (frequency) and then divide by the total number of individuals in that category.
For women, this involves multiplying 0 by 7350, 1 by 2587, and 2 by 80, then adding these results to get a total marriage frequency of 2747. The mean is found by dividing 2747 by the total number of women, which is 10017, resulting in a mean of approximately 0.274.
For men, the calculations are similar: 0 times 8418, 1 times 1594, and 2 times 10, adding up to 1614. Dividing this by the total number of men (10022) gives a mean of about 0.161.

These results show that, on average, the women in the dataset have been married more often than the men. The mean is useful here because it takes every data point into account, providing a comprehensive view.

Median Calculation

The median is the middle value in a list of numbers and is especially useful in skewed distributions to identify the typical outcome without being influenced by extreme values. Finding the median involves ranking all individuals by the number of times they were married and identifying the middle position.
For women, we start with the cumulative frequencies: the first category (never married) includes 7350 women. Adding the next group (married once), we reach 9937 women. Since the total is 10017, which is near the middle, the median falls in the category "1 time married."
For men, again starting with cumulative frequencies, there are 8418 that never married. Including the group married once brings us to 10012, very close to the total of 10022. So, the median is "0 times married."

While medians can downplay the frequency of multiple marriages by focusing on central values, in this exercise, it highlights that most men have never married, while a significant number of women have been married at least once.

Gender Comparison

Gender comparison in descriptive statistics often reveals social and cultural patterns and differences. In exercises like this, by calculating both the mean and median, we gain a rounded understanding of marriage patterns among different genders.
From our calculations, the mean number of marriages for women (0.274) is higher than for men (0.161). This suggests, on average, women in this age range have been married more often than men.
The medians tell a slightly different story. The median for women indicates they have married once, whereas for men, it reflects that most have never married. This suggests a more varied marriage experience among women, compared to a larger proportion of unmarried men.

Choosing between mean and median depends on the situation. The mean offers insight into the overall trend, while the median provides the typical experience, immune to extremes. In this analysis, both statistics are helpful: the mean illustrates average differences, whereas the median highlights the clustered experiences around fewer marriages among men.