91Ó°ÊÓ

In 2012, the General Social Survey asked respondents how many children they felt would be in an "ideal" family. The histogram contains the data from 1730 people who responded to the survey. a. Approximately what is the mean ideal number of children? Explain how you chose this value. b. What is the approximate value for the median ideal number of children? Describe how you chose this value. c. Find the mean by completing the work that is started below: $$ \bar{x}=\frac{18(0)+43(1)+965(2)+\cdots}{1730} $$ d. Explain how the method in part \(\mathrm{c}\) is related to the usual method of finding the mean, which has all the raw numbers given, without frequencies. e. Which is more appropriate to report for these data, the mean or the median? Why?

Step by step solution

01

Identifying Mean and Median

In a frequency distribution, the 'mean' refers to the average, obtained by summing all the values in the dataset and dividing by the number of values. The 'median' is the middle value when the data are sorted in numerical order. If there is an even number of observations, the median is the average of the two middle numbers.

02

Approximating the Mean

To approximate the mean ideal number of children, it is necessary to estimate the balance point of the histogram. It should be close to the most frequently reported number of children, which here is 2.

03

Approximating the Median

The median can be found by figuring out the middle value. Since there are 1730 observations, the middle would be around the 865th observation. Looking at the provided frequency distribution, we can see that the median is approximately 2.

04

Calculating the Actual Mean

To find the mean, multiply each score by its frequency, sum these products, and then divide the resulting sum by the total number of scores. The equation given in the problem is set up this way. The final answer will depend on the remaining summands—but for now, we can express the following: \(\bar{x}=\frac{sum of(numbers*frequency)}{1730}\).

05

Comparing the Methods for Mean Calculation

The technique used in part (c) is actually the standard method for finding the mean when frequencies are given. The key difference is that there's no need to explicitly list out every single individual score; instead, we use frequencies to reduce the calculation load.

06

Choosing Between Mean and Median

To decide which value is more appropriate to report, we should consider the skewness. If the number of children is approximately normally distributed, there would be little difference between mean and median, so either would be a reasonable measure. In a skewed distribution, the median could be more informative as it is less sensitive to extreme values.

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Key Concepts

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

Mean

In descriptive statistics, the mean is one of the most commonly used measures of central tendency. Think of the mean as a way to find the "average" value in a dataset. To calculate the mean, add up all the numbers and then divide by how many numbers there are.

When dealing with frequency distributions, like in the exercise, you multiply each distinct value by its frequency. This is what the equation from the original exercise depicts:

• Multiply each possible number of children by the number of people who think that number is ideal.
• Add up all these products.
• Divide that total by the number of respondents (1730 in this case).

This method of calculation is valuable because you don't need to write out each number separately, saving time and effort.

Median

The median is another key measure of central tendency, and it's especially useful when dealing with skewed data. The median is simply the middle number in a sorted list.

Here's how you find the median in a dataset:

• First, arrange the numbers in ascending order.
• If there's an odd number of observations, the median is the number right in the middle.
• If there's an even number of observations, the median is the mean of the two middle numbers.

In the case of the exercise, there are 1730 responses. So, to locate the median, you'd find the 865th number in the sorted sequence. It turns out to be 2 because this is where the cumulative count suggests the middle observation will fall.

Histogram

A histogram is a type of graph used to represent frequency distribution. It provides a visual interpretation of numerical data by indicating the number of data points that lie within a range of values, or "bins."

Histograms are useful because:

• They give a quick overview of the distribution and spread of your data.
• They are excellent at showing the shape of the data's distribution—whether it's skewed, bimodal, etc.

In the context of the exercise, the histogram depicts people's ideal number of children. Featuringly, it shows that the most common response is 2 children, which helps in understanding both the mean and the median.

Frequency Distribution

Frequency distribution is a tabular or graphical representation that displays how often each different value in a set of data occurs. This method gives clarity to complex datasets by displaying frequencies in a more organized form.

Key aspects include:

• Listing all possible outcomes on one axis.
• Showing the frequency of each outcome on the other axis.

The exercise you worked on uses frequency distribution to display how many people believe a certain number of children is ideal. By presenting the data in this manner, it becomes easier to calculate statistics like the mean or median, as the distribution pattern is clearly visible.