Question

The tuition and fees (in thousands of dollars) for a sample of 21 four-year state-run colleges and universities are shown in the following table. \(\begin{array}{rrrrrr}10.4 & 10.7 & 10.0 & 9.2 & 8.6 & 8.9 \\ 6.8 & 10.0 & 9.7 & 9.0 & 8.2 & \\ 13.0 & 6.8 & 6.5 & 6.4 & 12.0 & \\ 10.4 & 14.3 & 8.2 & 15.6 & 8.8 & \end{array}\) a. Find the mean, the median, and the mode. b. Compare the median and the mean. What can you say about the shape of this distribution? c. Draw a dotplot for the data. Does this confirm your conclusion about the shape of the distribution from part b?

Short answer

Answer: If the mean tuition is greater than the median tuition, the distribution is likely right-skewed. If the mean tuition is less than the median tuition, the distribution is likely left-skewed. If the mean tuition is equal to the median tuition, the distribution is likely symmetric.

Step-by-step solution

Sort the data

Arrange the tuition data in an ascending order to make it easier to work with and to find the median.

Find the mean

To find the mean, add up all the tuition values and divide the sum by the number of colleges (21).

Find the median

To find the median, locate the middle value of the sorted data. Since there are 21 colleges, the median will be the value in the 11th position.

Find the mode

To find the mode, look for the value(s) that occur most frequently in the data.

Compare the median and the mean

Observe the relationship between the mean and the median values and draw a conclusion about the shape of the distribution.

Draw a dotplot

Create a dotplot for the tuition data by placing a dot above a horizontal axis for each value. This will provide a visual representation of the data distribution.

Confirm your conclusion about the shape of the distribution

Compare the dotplot with the relationship between the mean and the median values, and check if it confirms your conclusion about the shape of the data distribution.

Key concepts

Mean, Median, and Mode

Understanding the measures of central tendency—mean, median, and mode—is essential for analyzing data sets in education. The mean is calculated by adding all the data points together and then dividing by the number of points. It gives a mathematical average and is sensitive to extremely high or low values, which can skew the mean.

The median is the middle value of an ordered data set, providing a measure that is not affected by outliers and thus often represents the data set more accurately in case of skewed distributions. For the given data set, where the tuition fees are ordered, the median would be the 11th value, as this is the center point of 21 sorted values.

The mode is the most frequently occurring value in the data set, which might indicate the most common fee charged by the state-run colleges. If there are no repeating values, the data set would have no mode. Knowing the mode can be particularly important when considering the most common cost scenario students might face.

When the mean and median are close to each other, the distribution can be considered fairly symmetrical. If they differ significantly, it suggests a skewed distribution. The calculation of these three measures for the college tuition sample can provide a comprehensive picture of the central tendencies in this educational data.

Data Distribution

The concept of data distribution refers to how values in a data set are spread out. In the context of educational statistics, understanding this spread can highlight key differences in costs, performance, or any other measurable variable between schools, programs, or students.

When analyzing the mean and median from the tuition fee example, if the mean is higher than the median, it indicates a right-skewed distribution, meaning some schools have much higher fees that push the mean up. Conversely, if the mean is lower, the distribution is left-skewed, with a few exceptionally low values affecting the average. A symmetrical distribution would see the mean and median very close in value.

Understanding the shape of a distribution helps educators and administrators identify outliers and make informed decisions based on the typical (median) or average (mean) experiences rather than extreme cases.

Dotplot

A dotplot is a simple visual tool used to display the distribution of a data set. It consists of a number line, and for each data point, a dot is placed above the line. In the context of the tuition data from state-run colleges, each tuition fee would be represented by a dot. When multiple colleges have the same fee, their dots stack vertically.

Creating a dotplot lets students visually assess the concentration of data points, the spread of the data, and the presence of outliers. For the given tuition fees, comparing the dotplot with the earlier calculations of mean and median can help confirm whether the data is skewed or symmetrical.

A right-skewed dotplot will show a pile-up of dots on the left with a few dots (representing the higher tuition fees) stretched out to the right. This visual representation is crucial as it enhances understanding of the data distribution beyond mere numerical analysis, making it an excellent tool for educational purposes.