Question

SOC The price of a gallon of regular gas for 20 nations is reported below. Compute the mean, median, range, interquartile range, and standard deviation for this variable, and write a paragraph summarizing these statistics.

Short answer

Mean, Median, Range, IQR, and Standard Deviation summarized.

Step-by-step solution

List the Prices

First, list the 20 different gas prices provided in the exercise.

Calculate the Mean

Add all the gas prices together and then divide the sum by 20 to find the mean. Formula: \[ \text{Mean} = \frac{\sum_{i=1}^{n} x_i}{n} \]

Find the Median

Organize the prices in ascending order. If the number of observations (n) is odd, the median is the middle number; if it is even, the median is the average of the two middle numbers.

Compute the Range

Subtract the smallest price from the largest price to get the range. Formula: \[ \text{Range} = \text{Max} - \text{Min} \]

Determine the Interquartile Range (IQR)

Find the first quartile (Q1) and the third quartile (Q3). Then, subtract Q1 from Q3. Formula: \[ \text{IQR} = Q3 - Q1 \]

Calculate the Standard Deviation

Use the formula for standard deviation: \[ s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2} \] where \( \bar{x} \) is the mean of the prices.

Summarize the Statistics

Write a paragraph summarizing the mean, median, range, interquartile range, and standard deviation of the gas prices.

Key concepts

mean calculation

The mean, or average, is a fundamental concept in descriptive statistics. It provides an indication of the central value of a set of numbers. To calculate the mean for the price of a gallon of regular gas from 20 nations, follow these steps: add together all the given prices to get the total sum. Then, divide this sum by the number of prices (which is 20 in this case). The formula looks like this: \( \text{Mean} = \frac{\text{sum of all values}}{\text{number of values}} \). This calculation helps you understand the typical price of gas across these nations.

median determination

The median value is the midpoint number in a sorted list of numbers. To find the median, first arrange all 20 gas prices in ascending order. If the number of observations \( n \) is odd, the median is the middle value. But if \( n \) is even, it is the average of the two middle numbers. For example, with our 20 gas prices, we need to take the 10th and 11th prices in the sorted list and average them to get the median. This measure reduces the impact of outliers and gives a better central tendency for skewed distributions.

range computation

The range is a simple measure of variability that provides the spread between the highest and the lowest values. To compute the range, subtract the smallest price from the largest price using the formula: \( \text{Range} = \text{Max} - \text{Min} \). This will give you an idea of how varied the gas prices are across the 20 nations. Although easy to calculate, the range can be influenced greatly by outliers.

interquartile range

The interquartile range (IQR) offers a more refined measure of variability by focusing on the spread of the middle 50% of data. To find the IQR, first calculate the first quartile \( Q1 \) and the third quartile \( Q3 \). Quartiles divide data into four equal parts. Specifically, \( Q1 \) is the median of the lower half of the data set and \( Q3 \) is the median of the upper half. Subtract \( Q1 \) from \( Q3 \) to get the IQR, using the formula: \( \text{IQR} = Q3 - Q1 \). The IQR provides insight into the spread of the central bulk of data, reducing the influence of extreme values.

standard deviation

Standard deviation is a widely-used measure to quantify the amount of variation or dispersion of a set of values. To compute the standard deviation, follow these steps:

• Calculate the mean of the data set.
• Subtract the mean from each data value to find the deviation for each value.
• Square these deviations.
• Find the average of these squared deviations.
• Finally, take the square root of this average to get the standard deviation.

The formula is: \( s = \frac{1}{n-1} \root{2} \sum_{i=1}^{n} (x_i - \bar{x})^2} \). Standard deviation gives a broader perspective on the spread of data around the mean.