Math Problem Statement

Listed below are the annual tuition amounts of the 10 most expensive colleges in a country for a recent year. What does this​ "Top 10" list tell us about the population of all of that​ country's college​ tuitions? $ 54,268  $ 52,820  $ 54,298  $ 52,678  $ 52,319
$ 52,561  $ 54,219  $ 52,820  $ 53,772  $ 50,963 Question content area bottom Part 1 Find the​ mean, midrange,​ median, and mode of the data set. The mean of the data set is ​$   

​Round to two decimal places as​ needed.

Solution

Let's first calculate the required statistical measures: the mean, midrange, median, and mode of the given tuition amounts.

1. Mean

The mean is calculated by summing all the values and dividing by the number of values.

Given dataLet's first calculate the required statistical measures: the mean, midrange, median, and mode of the given tuition amounts.

1. Mean

The mean is calculated by summing all the values and dividing by the number of values.

Given Data Set={54,268,52,820,54,298,52,678,52,319,52,561,54,219,52,820,53,772,50,963}\text{Data Set} = \{54,268, 52,820, 54,298, 52,678, 52,319, 52,561, 54,219, 52,820, 53,772, 50,963\}

Sum of the values: 54,268+52,820+54,298+52,678+52,319+52,561+54,219+52,820+53,772+50,963=530,71854,268 + 52,820 + 54,298 + 52,678 + 52,319 + 52,561 + 54,219 + 52,820 + 53,772 + 50,963 = 530,718

Number of values: n=10n = 10

Mean: Mean=530,71810=53,071.80\text{Mean} = \frac{530,718}{10} = 53,071.80

2. Midrange

The midrange is calculated as the average of the maximum and minimum values in the data set.

Minimum value: Min=50,963\text{Min} = 50,963

Maximum value: Max=54,298\text{Max} = 54,298

Midrange: Midrange=Min+Max2=50,963+54,2982=105,2612=52,630.50\text{Midrange} = \frac{\text{Min} + \text{Max}}{2} = \frac{50,963 + 54,298}{2} = \frac{105,261}{2} = 52,630.50

3. Median

The median is the middle value when the data set is ordered. Since there are 10 values, the median will be the average of the 5th and 6th values.

Ordered data set: 50,963,52,319,52,561,52,678,52,820,52,820,53,772,54,219,54,268,54,29850,963, 52,319, 52,561, 52,678, 52,820, 52,820, 53,772, 54,219, 54,268, 54,298

Median: Median=52,820+52,8202=52,820\text{Median} = \frac{52,820 + 52,820}{2} = 52,820

4. Mode

The mode is the value that occurs most frequently in the data set.

In this case, the value 52,820 appears twice, so the mode is: Mode=52,820\text{Mode} = 52,820

Summary of Results

  • Mean: 53,071.8053,071.80
  • Midrange: 52,630.5052,630.50
  • Median: 52,82052,820
  • Mode: 52,82052,820

This "Top 10" list provides insight into the higher end of the tuition spectrum but does not give a complete picture of the overall distribution of college tuition costs in the country. The measures suggest that tuition amounts among the most expensive colleges are relatively close, but they are likely not representative of the entire population of tuition costs, which could include much lower figures.


Would you like further details on any of these calculations? Here are some related questions:

  1. How does the median compare to the mean in this data set?
  2. What does the presence of a mode tell us about the data distribution?
  3. How would the midrange change if we included lower-cost colleges?
  4. What could be the potential effects of outliers on these statistics?
  5. How would you interpret these results if more data points were available?

Tip: When analyzing data, it's crucial to understand whether the data set is representative of the entire population, especially when dealing with extreme values like top 10 lists.

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Math Problem Analysis

Mathematical Concepts

Statistics
Descriptive Statistics
Measures of Central Tendency

Formulas

Mean = (Sum of values) / (Number of values)
Midrange = (Min value + Max value) / 2
Median = Middle value (or average of two middle values) in an ordered data set
Mode = Most frequent value

Theorems

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Suitable Grade Level

Grades 9-12