Question

a meteorologist is studying the monthly rainfall in a section of the brazilian rainforest. she recorded the monthly rainfall, in inches, for last year. they were:

10.4, 10.3, 11.7, 11.1, 8.0, 4.4, 2.6, 1.8, 2.5, 4.4, 7.3, 9.5

calculate the following for the data set:
mean
median
mode
range

which of the following is true if the data point 11.1 is corrected to 11.7?

• the mean increases
• the median increases
• the data set has two modes
• the range increases

if the next months rainfall is 23.1 inches, this will have the biggest effect on the mean / median / mode

Explanation:

Analyze the original data set

Using the Basic Statistics knowledge point
The original data set consists of 12 monthly rainfall values:

$$ 10.4, 10.3, 11.7, 11.1, 8.0, 4.4, 2.6, 1.8, 2.5, 4.4, 7.3, 9.5 $$

Sorted in ascending order:

$$ 1.8, 2.5, 2.6, 4.4, 4.4, 7.3, 8.0, 9.5, 10.3, 10.4, 11.1, 11.7 $$

• Mean: \(\frac{1.8 + 2.5 + 2.6 + 4.4 + 4.4 + 7.3 + 8.0 + 9.5 + 10.3 + 10.4 + 11.1 + 11.7}{12} = \frac{84}{12} = 7\)
• Median: Average of the 6th and 7th values: \(\frac{7.3 + 8.0}{2} = 7.65\)
• Mode: \(4.4\) (appears twice)
• Range: \(11.7 - 1.8 = 9.9\)

Analyze the effect of correcting 11.1 to 11.7

Using the Basic Statistics knowledge point
The data point \(11.1\) is corrected to \(11.7\).
The new sorted data set is:

$$ 1.8, 2.5, 2.6, 4.4, 4.4, 7.3, 8.0, 9.5, 10.3, 10.4, 11.7, 11.7 $$

• Mean: The sum increases by \(0.6\), so the mean increases to \(\frac{84.6}{12} = 7.05\). (True)
• Median: The 6th and 7th values remain \(7.3\) and \(8.0\), so the median remains \(7.65\). (False)
• Mode: Both \(4.4\) and \(11.7\) now appear twice. The data set is bimodal, meaning it has two modes. (True)
• Range: The maximum value remains \(11.7\) and the minimum remains \(1.8\), so the range remains \(9.9\). (False)

Analyze the effect of adding an outlier

Using the Outlier Effects on Statistics knowledge point
An additional data point of \(23.1\) inches is introduced. This value is significantly larger than the rest of the data set (where the maximum was \(11.7\)), making it a high outlier.

• Mean: Highly sensitive to outliers because every value is included in the sum. Adding \(23.1\) will pull the mean significantly upward.
• Median: Highly resistant to outliers because it only depends on the middle position of the ordered list.
• Mode: Completely unaffected by a single extreme value unless it changes the most frequent value.

Therefore, adding \(23.1\) has the biggest effect on the mean.

Answer:

Question 1

Calculate the following for the data set:

• Mean: 7
• Median: 7.65
• Mode: 4.4
• Range: 9.9

Question 2

Which of the following is true if the data point 11.1 is corrected to 11.7?

• The mean increases
• The data set has two modes

Question 3

If the next month's rainfall is 23.1 inches, this will have the biggest effect on the mean.