Question

A group of 70 students were asked to record the last digit of their social security number.a. Draw a relative frequency histogram using the values 0 through 9 as the class midpoints. What is the shape of the distribution? Based on the shape. what would be your best estimate for the mean of the data set? b. Use the range approximation to guess the value of \(s\) for this set. c. Use your calculator to find the actual values of \(\bar{x}\) and \(s\). Compare with your estimates in parts a and \(\mathbf{b}\).

Short answer

Question: Based on the steps provided, describe how you would create a relative frequency histogram, estimate the mean, and approximate the value of the standard deviation for a given group of 70 students with the last digit of their social security numbers as data points. Also, discuss how to calculate the actual mean and standard deviation and compare them with the estimations.

Step-by-step solution

Organizing the data

Using the data provided, create a frequency table that shows how many students have a social security number ending in each digit from 0 to 9. Remember to record both the count and relative frequency.

Draw a relative frequency histogram

Using the relative frequencies calculated in step 1, draw a histogram to display the distribution of values. The class midpoints will be the integers from 0 to 9, representing the last digit of each student's social security number.

Describe the shape of the distribution and estimate the mean

Observe the histogram to determine the shape of the distribution. Depending on the distribution, your best estimate for the mean will vary. For a symmetric distribution, the mean will be in the center. For a skewed distribution, the mean will be slightly shifted towards the tail.

Use range approximation to guess the value of s

To estimate the standard deviation (s) using the range approximation, first calculate the range, which is the difference between the maximum and minimum values. Then, divide the range by 4 (a rough estimate for the number of 'steps' from the lowest to the highest value) to approximate the standard deviation (s): \(s \approx \frac{range}{4}\)

Calculate the actual values of \(\bar{x}\) and \(s\) using a calculator

Enter the data into your calculator or use a software tool to find the actual mean (\(\bar{x}\)) and standard deviation (s) for the dataset.

Compare your estimations with the actual values

Compare the estimated mean and standard deviation from parts a and b with the actual values calculated in step 5. Note the differences between your estimates and the actual values, and discuss the accuracy of your estimations.

Key concepts

Relative Frequency Histogram

A relative frequency histogram is a visual tool used to display the distribution of data points by showing the relative frequency of each possible value. In the given exercise, the last digits of students' social security numbers are recorded, with values ranging from 0-9. To create this histogram, first construct a frequency table recording how often each digit appears. Then, convert these frequencies to relative frequencies by dividing each by the total number of observations, which in this case is 70. This way, you find out not just how often each number appears, but how significant its appearance is relative to the overall data set.
This type of histogram is particularly useful because it shows the proportions directly on a 0-1 scale, making it easier to compare different distributions. You can quickly observe if some digits occur significantly more often than others, or if they appear with equal likelihood. This visualization forms the foundational step for further analysis like mean estimation or distribution shape analysis.

Mean Estimation

To estimate the mean of a dataset, especially when you have a graphical representation like a histogram, you can look at the shape of the distribution. The mean is the "average" value of the dataset and can be roughly estimated by identifying the central tendency of the histogram. For a symmetric histogram, the mean will likely be at or near the center of the distribution.
When a distribution is skewed, the mean will be pulled towards the tail, which is the longer part of the histogram. For example, if the histogram tail extends more on the right, the mean will likely be higher than the median. In many cases, estimating the mean visually can help inform whether formal calculation is necessary or to provide a check against detailed numerical analysis. In this exercise, comparing visual estimates of the mean with those calculated by a calculator helps validate the graphical approach.

Standard Deviation Approximation

Standard deviation is a crucial measure in statistics, indicating how spread out the numbers in a dataset are. It tells us how much variation or "dispersion" from the mean exists. To approximate the standard deviation using the range rule of thumb, you first need to calculate the range of the dataset, which is the difference between the maximum and minimum values.
With the range known, the estimate for the standard deviation is obtained by dividing the range by 4. This method provides a quick, rough estimate that is particularly useful when you need a general sense of the data's variability but don't require exact precision. However, it's important to note that this approximation may not be as accurate for non-symmetrical or multi-modal distributions, and should be compared to actual calculations for confirmation.

Distribution Shape Analysis

The shape of a distribution reveals a lot about how data points are spread across different values. In this exercise, analyzing the shape of the histogram can help identify whether the distribution is normal, skewed, or has any other distinctive patterns. A normal distribution will typically look like a "bell curve," symmetric with most values clustering around the mean.
Skewness influences our understanding of both mean and standard deviation, as data skewed to the left (negative skew) indicates a longer left tail, while a right-skewed (positive skew) distribution has a longer right tail. Additionally, the presence of a bimodal distribution, where the histogram shows two prominent peaks, suggests that the dataset may be influenced by two different processes or groups.

• Symmetric distribution: Mean approximates the median.
• Positively skewed: Mean is greater than median.
• Negatively skewed: Mean is less than median.

Patterns recognized in distribution shape analysis lend essential insight into data analysis, guiding interpretations of further statistical measures.