Question

The table below shows the names of the 43 presidents of the United States along with the number of their children. \(^{9}\) a. Construct a relative frequency histogram to describe the data. How would you describe the shape of this distribution? b. Calculate the mean and the standard deviation for the data set. c. Construct the intervals \(\bar{x} \pm s, \bar{x} \pm 2 s,\) and \(\bar{x} \pm 3 s\). Find the percentage of measurements falling into these three intervals and compare with the corresponding percentages given by Tchebysheff's Theorem and the Empirical Rule.

Short answer

Question: Based on the constructed intervals, calculate the percentage of U.S. presidents' number of children falling within 1, 2, and 3 standard deviations of the mean, and compare the results with Tchebysheff's Theorem and the Empirical Rule.

Step-by-step solution

Create a Relative Frequency Histogram

First, we need to organize the data into classes or bins. We must count the frequency of each class and compute the relative frequency for each class by dividing the frequency by the total number of presidents (43). Use a graphing tool or spreadsheet software to create the histogram, with the number of children on the x-axis and the relative frequency on the y-axis. Analyze the histogram to describe the shape of the distribution – whether it is symmetric, skewed, or uniform.

Calculate the Mean (Average)

To calculate the mean (\(\bar{x}\)) of the data set, add up all the number of children for each president and divide by the total number of presidents (43). \(\bar{x} = \frac{\Sigma x}{n}\), where \(x\) is the number of children and \(n\) is the total number of data points (43).

Calculate the Standard Deviation

To calculate the standard deviation (\(s\)), we will use the formula \(s = \sqrt{\frac{\Sigma(x - \bar{x})^2}{n-1}}\), where \(x\) is each data point (number of children), \(\bar{x}\) is the mean calculated in Step 2, and \(n\) is the total number of data points (43). Compute the squared deviations, add them up, divide the sum by 42 (n-1), and take the square root of the result.

Construct the Intervals

Next, we need to construct the intervals centered at the mean, as follows: 1. \(\bar{x} \pm s\) (Mean \(\pm\) Standard Deviation) 2. \(\bar{x} \pm 2s\) (Mean \(\pm\) 2 × Standard Deviation) 3. \(\bar{x} \pm 3s\) (Mean \(\pm\) 3 × Standard Deviation)

Calculate the Percentage of Measurements in Each Interval

We will now calculate the percentage of measurements falling into the intervals calculated in Step 4, by counting the number of presidents whose number of children falls within each interval and dividing that by the total number of presidents (43).

Compare with Tchebysheff's Theorem and the Empirical Rule

Finally, we will compare the results obtained in Step 5 with the corresponding percentages given by Tchebysheff's Theorem and the Empirical Rule. Tchebysheff's Theorem states that at least \((1 - \frac{1}{k^2})\) of the data should fall within \(k\) standard deviations of the mean. The Empirical Rule states that approximately 68%, 95%, and 99.7% of the data should fall within 1, 2, and 3 standard deviations of the mean, respectively, for a symmetric, unimodal distribution. Compare the intervals' actual percentages from Step 5 with these theoretical percentages.

Key concepts

Relative Frequency Histogram

A relative frequency histogram is a visual representation of data, showing the relative frequencies of different categories or classes. In this context, the x-axis represents the number of children each U.S. president had, while the y-axis shows the relative frequency, which is the frequency of a class divided by the total number of presidents (43 in this case).
To construct a relative frequency histogram, you need to first organize the data into classes. For example, if you're counting the number of children, you might decide on classes such as 0-2 children, 3-5 children, etc.
Once classes are defined, count how many presidents fall into each class and divide by the total number of presidents to get the relative frequency. Plot these frequencies as bars on your graph, where the height of each bar represents the relative frequency.
This type of histogram can help you quickly assess the distribution shape:

• If it is symmetric, it means the data is evenly spread around the mean.
• A skewed distribution indicates data is piled up on one end.
• An even distribution, or uniform shape, means all data points are equally likely.

By examining the shape, one can glean insights into the dataset's characteristics and underlying patterns.

Mean and Standard Deviation

The mean, or average, gives us a central value for a dataset. To compute the mean (\(\bar{x}\)), add together all the values (in our case, the number of children for each president) and divide by the total number of data points, which is 43 presidents.
The formula used is:
\[\bar{x} = \frac{\Sigma x}{n}\]
where \(\Sigma x\) is the sum of all data points, and \(n\) is the number of data points.
The standard deviation (\(s\)) provides a measure of the data's spread or dispersion from the mean. A smaller standard deviation indicates data points are close to the mean, while a larger one indicates data is spread out over a larger range of values.
The formula for standard deviation is:
\[s = \sqrt{\frac{\Sigma(x - \bar{x})^2}{n-1}}\]
Here, \(x\) represents each data point, \(\bar{x}\) is the mean, and \(n\) is the number of data points. Calculating it involves determining the squared difference of each data point from the mean, summing all squared differences, dividing by 42 (one less than the number of data points), and then taking the square root.
These two measures, mean and standard deviation, together provide a comprehensive understanding of the dataset's central tendency and variability.

Tchebysheff's Theorem

Tchebysheff's theorem offers a way to estimate the spread of data points around the mean, applicable to any distribution shape. It states that at least \((1 - \frac{1}{k^2})\) of the data falls within \(k\) standard deviations from the mean, where \(k\) is any number greater than 1. This theorem is valuable as it provides a guarantee applicable to all distributions, unlike the Empirical Rule, which assumes normality.
In practical terms:

• For \(k = 2\), at least 75% of data should lie within 2 standard deviations of the mean.
• For \(k = 3\), at least 88.9% falls within 3 standard deviations.

Tchebysheff's theorem is especially useful when little is known about the data's distribution shape.
However, it is a lower bound theorem, meaning the actual percentage of data falling within these ranges can be higher. When comparing the percentages of data within calculated intervals against Tchebysheff's predictions, one can assess the concentration and spread of data more objectively.

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, applies specifically to normal (bell-shaped) distributions. It stipulates that:

• About 68% of the data falls within one standard deviation of the mean \((\bar{x} \pm s)\).
• Approximately 95% is within two standard deviations \((\bar{x} \pm 2s)\).
• Roughly 99.7% lies within three standard deviations \((\bar{x} \pm 3s)\).

This rule allows for a quick way to understand the spread of data and is particularly powerful in stats, as many natural phenomena follow a normal distribution.
While using the Empirical Rule, it's important to verify if the data closely follows a normal distribution. In contrast to Tchebysheff's theorem, which applies to any distribution, the Empirical Rule assumes this normality.
By comparing empirical results with this rule, one can validate how well the data aligns with a normal distribution. A significant difference suggests that the data might not be normally distributed. This insight is crucial when using statistical tools relying on normality assumptions.