Question

A variety of information has been collected for all district high schools. Find the most appropriate measure of central tendency for each variable and summarize this information in a paragraph. (HINT: The level of measurement of the variable will generally tell you which measure of central tendency is appropriate. Remember to organize the scores from high to low before finding the median.) $$\begin{array}{cccccc} \text {High School } & \text { Enrollment } & \text {Largest Racial/ Ethnic Group } & \text {Percent College Bound } & \text {Most Popular Sport } & \text {Condition of Physical Plant (scale of \(1-10\) with } 10 \text { high }) \\ \hline 1 & 1400 & \text { White } & 25 & \text { Football } & 10 \\ 2 & 1223 & \text { White } & 77 & \text { Baseball } & 7 \\ 3 & 876 & \text { Black } & 52 & \text { Football } & 5 \\ 4 & 1567 & \text { Hispanic } & 29 & \text { Football } & 8 \\ 5 & 778 & \text { White } & 43 & \text { Basketball } & 4 \\ 6 & 1690 & \text { Black } & 35 & \text { Basketball } & 5 \\ 7 & 1250 & \text { White } & 66 & \text { Soccer } & 6 \\ 8 & 970 & \text { White } & 54 & \text { Football } & 9 \end{array}$$

Short answer

Median enrollment: 1236.5. Mode for largest racial/ethnic group: White. Median percent college bound: 47.5%. Mode for most popular sport: Football. Median condition of physical plant: 6.5.

Step-by-step solution

Identify the Level of Measurement

Determine the level of measurement for each variable:- Enrollment: Ratio- Largest Racial/Ethnic Group: Nominal- Percent College Bound: Ratio- Most Popular Sport: Nominal- Condition of Physical Plant: Ordinal

Choose Measures of Central Tendency

For each variable, choose the appropriate measure of central tendency based on its level of measurement:- Enrollment: Mean or median- Largest Racial/Ethnic Group: Mode- Percent College Bound: Mean or median- Most Popular Sport: Mode- Condition of Physical Plant: Median

Gather Data and Organize

Organize the data from the table:- Enrollment: {1400, 1223, 876, 1567, 778, 1690, 1250, 970}- Largest Racial/Ethnic Group: {White, White, Black, Hispanic, White, Black, White, White}- Percent College Bound: {25, 77, 52, 29, 43, 35, 66, 54}- Most Popular Sport: {Football, Baseball, Football, Football, Basketball, Basketball, Soccer, Football}- Condition of Physical Plant: {10, 7, 5, 8, 4, 5, 6, 9}

Find Measures of Central Tendency

Calculate or find the appropriate measure of central tendency for each variable:- Enrollment: - Ordered: {778, 876, 970, 1223, 1250, 1400, 1567, 1690} - Median: (1223 + 1250) / 2 = 1236.5- Largest Racial/Ethnic Group: - Mode: White- Percent College Bound: - Ordered: {25, 29, 35, 43, 52, 54, 66, 77} - Median: (43 + 52) / 2 = 47.5- Most Popular Sport: - Mode: Football- Condition of Physical Plant: - Ordered: {4, 5, 5, 6, 7, 8, 9, 10} - Median: (6 + 7) / 2 = 6.5

Summarize Findings

Summarize the findings in a paragraph:The median enrollment in district high schools is 1236.5 students. The most common largest racial/ethnic group is White. The median percent of students that are college bound is 47.5%. Football is the most popular sport among the schools. The median condition of the physical plants, on a scale of 1-10, is 6.5.

Key concepts

levels of measurement

Understanding the levels of measurement is key to selecting the right statistical methods. There are four main levels of measurement: nominal, ordinal, interval, and ratio. Each level tells us about the nature of the data and how we can handle it.
Nominal data categorizes without a quantitative value, such as racial groups or types of sports. Ordinal data has a natural order, like the condition of the physical plant rated from 1 to 10.
Interval data includes equal intervals between values but no true zero point, like temperature scales (though it's less common in educational statistics). Lastly, ratio data has both equal intervals and a true zero. This includes enrollment numbers and percentages of college-bound students.
Knowing these distinctions helps us choose the right measures of central tendency for our data.

variable categorization

Categorizing variables depends on their nature and the type of data they represent. Here are the main types:

• Nominal variables: These are categorical and include data like the largest racial/ethnic group and the most popular sport. They don't have a specific order or quantitative value but rather serve to categorize.
• Ordinal variables: These variables have a specific order but the intervals between values aren't equal. The condition of the physical plant, rated on a scale from 1 to 10, is an example.
• Interval variables: Although not used in our example, interval variables have equal intervals between values with no true zero, like the temperature in Celsius.
• Ratio variables: These have both equal intervals and a true zero. Examples include enrollment numbers and the percentage of college-bound students.

Accurate categorization of variables ensures that we choose the appropriate statistical methods for analysis.

median calculation

The median is a measure of central tendency that represents the middle value in an ordered data set. To find the median, follow these steps:

• Step 1: Arrange the data in ascending order.
• Step 2: If the number of observations is odd, the median is the middle value.
• Step 3: If the number is even, the median is the average of the two middle numbers.

For example, consider the enrollment numbers: {1400, 1223, 876, 1567, 778, 1690, 1250, 970}. When ordered, they become {778, 876, 970, 1223, 1250, 1400, 1567, 1690}. Since there are eight numbers, the median is (1223 + 1250) / 2 = 1236.5. This method ensures we find the value that best represents the center of our data set.

mode identification

The mode is the value that appears most frequently in a data set. It serves as another measure of central tendency, particularly useful for nominal data. Finding the mode involves straightforward steps:

• Step 1: List all values in the data set.
• Step 2: Count the frequency of each value.
• Step 3: Identify the value with the highest frequency.

For example, in the exercise the data for the most popular sport is: {Football, Baseball, Football, Football, Basketball, Basketball, Soccer, Football}. Football appears the most, making it the mode. Similarly, for the largest racial/ethnic group data set, White appears frequently, thus it's the mode. Identifying the mode helps us understand the most common values in nominal data.

data organization

Effective data organization forms the backbone of accurate statistical analysis. Organizing data involves:

• Clearing and structuring raw data into meaningful formats.
• Arranging values in ascending or descending order to facilitate calculation of metrics like the median.
• Classifying and categorizing data based on variable types, as seen in the original problem.

By organizing data, we ensure that calculations are easier and analyses are more reliable. For instance, enrollment numbers must be ordered to find the median, while categorical data like the most popular sport should be summarized to identify the mode. Proper data organization simplifies finding measures of central tendency, making it crucial for informed decision-making.