Question

A meteorologist preparing a talk about global warming compiled a list of weekly low temperatures (in degrees Fahrenheit) he observed at his southern Florida home last year. The coldest temperature for any week was \(36^{\circ} \mathrm{F}\), but he inadvertently recorded the Celsius value of \(2^{\circ}\). Assuming that he correctly listed all the other temperatures, explain how this error will affect these summary statistics: a) measures of center: mean and median. b) measures of spread: range, IQR, and standard deviation.

Short answer

The mean and range are slightly higher; the median and IQR remain unchanged. The standard deviation is slightly smaller due to the error.

Step-by-step solution

Convert Celsius to Fahrenheit

First, convert the Celsius temperature back to Fahrenheit. The formula for this conversion is: \[ T(^{\circ}F) = \frac{9}{5} \times T(^{\circ}C) + 32 \]Plugging in the values, we have: \[ T(^{\circ}F) = \frac{9}{5} \times 2 + 32 = 35.6^{\circ}F \]So, the correct Fahrenheit temperature should have been approximately \(35.6^{\circ}F\).

Compare Recorded Versus Correct Value

Compare the wrongly recorded Fahrenheit value (\(36^{\circ}F\)) with the correct converted value (\(35.6^{\circ}F\)). The original incorrect entry was slightly higher than the actual temperature.

Impact on Measures of Center: Mean

The mean is calculated by summing all the values and dividing by the number of values. Due to the small error from \(36^{\circ}F\) to \(35.6^{\circ}F\), the mean will be slightly higher than it should be, but since the difference is small, the impact on the mean is negligible.

Impact on Measures of Center: Median

The median is the middle value when all observations are ordered. Since this temperature error affected only the value that was already the minimum, and likely did not change its position as the minimum value, the median is unchanged.

Impact on Measures of Spread: Range

The range is the difference between the maximum and minimum values. Since the minimum value is slightly higher than it should be, the calculated range (using the wrong entry: \(36^{\circ}F\)) will be slightly smaller.

Impact on Measures of Spread: IQR

The Interquartile Range (IQR) depends on the middle 50% of the data, not on extreme values such as the minimum or maximum. Hence, the IQR is unaffected by this error.

Impact on Measures of Spread: Standard Deviation

Standard deviation measures the spread of values around the mean. The incorrect minimum value is slightly higher, contributing to a slightly decreased spread, hence a slightly smaller standard deviation.

Key concepts

Mean

The mean is the average of a set of numbers. You calculate it by adding up all the numbers and then dividing by how many numbers there are. In the meteorologist's temperature data, an error occurred in recording the temperature. Instead of the actual temperature of approximately \(35.6^{\circ}F\), a slightly higher value of \(36^{\circ}F\) was used. Since the mean includes all values in its calculation, this small recording error causes the mean to be slightly larger than if the actual temperature had been used. The difference is minimal and might not drastically change the mean, but precision is key in statistics.

• Add all temperatures together.
• Divide by the number of temperature readings.
• Notice a higher value can increase the mean slightly.

Median

The median represents the middle value in a data set when it's arranged in order. It's as simple as lining up your numbers from smallest to largest and finding the one in the middle. If the data set has an even number of entries, the median will be the average of the two middle numbers. In our provided scenario, even with the wrong temperature being used, the median remains unaffected. This is because the error only alters the smallest value and does not change the position of the middle value(s).

• Line up all temperatures from lowest to highest.
• Identify the middle value.
• Realize the minimum value error doesn't shift the median.

Standard Deviation

Standard deviation is a way to understand how much your data set varies from the mean. It reveals how spread out the numbers are. If numbers in a list are close to the mean, the standard deviation is low; if they are spread out, it's high. In the example, the slightly elevated recorded minimum temperature reduces overall data variance, resulting in a slightly smaller standard deviation. This is because the deviation of each value from the mean slightly decreases when one value increases.

• Calculate the mean.
• Find how each number differs from the mean.
• Average the differences to find variance, then take the square root.
• Notice a smaller range can lead to a lower standard deviation.

Range

The range is one of the simplest measures of spread. You find it by subtracting the smallest value from the largest in your data set. In this case, with the smallest temperature misrecorded as slightly higher than it should be, the range becomes a bit smaller. This error reduces the difference between the highest and lowest temperatures in the dataset. Nonetheless, the impact of this minor change in range is typically not significant, but still, precision matters in accurate statistical analysis.

• Note the largest and smallest values in the list.
• Subtract the smallest from the largest.
• See how a larger minimum decreases the range.

Interquartile Range (IQR)

The Interquartile Range (IQR) focuses on the middle 50% of your data, making it a helpful statistic for understanding central dispersion. To compute the IQR, locate the first quartile (Q1), which is the median of the first half of the data, and the third quartile (Q3), the median of the second half. Subtract Q1 from Q3 to find the IQR. In the meteorologist’s data, the minimum temperature's error doesn’t affect the middle data, thus the IQR remains unchanged.

• Find Q1 and Q3 in sorted data.
• Subtract Q1 from Q3 to determine IQR.
• The unaffected middle range keeps IQR steady.