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Chapter 3: Problem 1

The data show the heights in feet of 14 roller coasters. Find the mean, median, midrange, and mode for the data $$\begin{array}{rrrrrrr}95 & 105 & 50 & 125 & 102 & 120 & 160 \\\102 & 118 & 91 & 160 & 95 & 50 & 84\end{array}$$

Short Answer

Expert verified
Mean: 96.93 feet, Median: 102 feet, Midrange: 105 feet, Mode: 50, 95, 102, and 160.

Step by step solution

01

Arrange in Order

First, organize the data in ascending order: 50, 50, 84, 91, 95, 95, 102, 102, 105, 118, 120, 125, 160, 160.
02

Calculate the Mean

Add all the heights together: 50 + 50 + 84 + 91 + 95 + 95 + 102 + 102 + 105 + 118 + 120 + 125 + 160 + 160 = 1357. Then divide the total by the number of roller coasters, which is 14. \[ \text{Mean} = \frac{1357}{14} \approx 96.93 \text{ feet} \]
03

Find the Median

With 14 numbers, the median will be the average of the 7th and 8th values in the ordered list: 102 and 102. \[ \text{Median} = \frac{102 + 102}{2} = 102 \text{ feet} \]
04

Calculate the Midrange

Add the smallest and largest values, then divide by 2: \[ \text{Midrange} = \frac{50 + 160}{2} = 105 \text{ feet} \]
05

Identify the Mode

The mode is the number that appears most frequently. In the dataset, 50, 95, 102, and 160 each appear twice. Hence, there are multiple modes: 50, 95, 102, and 160.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation
To find the mean of a dataset, you simply add up all the individual data points and then divide by the total number of points. In this exercise, the heights of 14 roller coasters are provided. First, sum all these heights, \[ 50 + 50 + 84 + 91 + 95 + 95 + 102 + 102 + 105 + 118 + 120 + 125 + 160 + 160 = 1357 \]. Next, divide this sum by the number of heights, which is 14: \[ \text{Mean} = \frac{1357}{14} \approx 96.93 \text{ feet} \].
  • Mean is a measure of central tendency, providing an average height.
  • Useful in understanding general expectations or predictions.
  • Every value affects the mean due to the addition of all data points.
Median Calculation
The median is another form of average, which requires arranging the data in ascending order. It represents the middle point of a dataset. For these 14 roller coaster heights, arrange them as follows: 50, 50, 84, 91, 95, 95, 102, 102, 105, 118, 120, 125, 160, 160. Since there are an even number of data points, the median is the average of the two middle numbers, which are 102 and 102:\[ \text{Median} = \frac{102 + 102}{2} = 102 \text{ feet} \].
  • In an ordered dataset, half the numbers are below the median and half are above.
  • Less affected by extremely high or low values, compared to the mean.
  • Gives a better idea of the "central" tendency when data distribution is skewed.
Midrange Calculation
The midrange is calculated by taking the sum of the smallest and largest values in a dataset and dividing it by two. This measure gives a sense of the average between the extremes. In this dataset of roller coaster heights, the smallest value is 50 and the largest is 160. To find the midrange:\[ \text{Midrange} = \frac{50 + 160}{2} = 105 \text{ feet} \].
  • Midrange helps understand the spread and center of the data.
  • Can be misleading if outliers are present, as it depends heavily on the extremes.
  • Simple to compute, but not often used as the sole measure of central tendency.
Mode Identification
The mode is defined as the number that appears most frequently in a dataset. It is possible for data to have more than one mode or even no mode at all. In this problem, the roller coaster heights 50, 95, 102, and 160 each occur twice, making them all modes of the dataset.
  • Mode is the only measure of central tendency that can be used with nominal data.
  • Useful for categorical data analysis, providing the most common category or item.
  • Can be more than one mode (multimodal) or no mode (all values unique).

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