Consider the distribution below. a) From the distribution, obtain the five number summary. b) What is the (i) mode and the (ii) midrange of the distribution? c) Calculate the mean... Consider the distribution below. a) From the distribution, obtain the five number summary. b) What is the (i) mode and the (ii) midrange of the distribution? c) Calculate the mean using the relationship existing among mode, median, mean.
Understand the Problem
The question is asking to analyze a distribution of data by obtaining the five-number summary, determining the mode and midrange, and calculating the mean based on the relationship among mode, median, and mean. This involves statistical concepts.
Answer
Minimum: $1$, Q1: $5$, Median: $22$, Q3: $223$, Maximum: $3556788999$, Mode: $5$, Midhinge: $114$, Mean: $17.75$.
Answer for screen readers
The five-number summary is:
- Minimum: $1$
- Q1: $5$
- Median: $22$
- Q3: $223$
- Maximum: $3556788999$
Mode: $5$
Midhinge: $114$
Mean: $17.75$
Steps to Solve
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Organize the Data First, we need to list the data in ascending order to facilitate calculations. The given data is: $$ 1, 1, 1, 2, 2, 5, 5, 5, 6, 8, 10, 11, 17, 18, 26, 32, 38, 45, 56, 64, 78, 123, 145, 223, 355, 478, 1234, 1234, 5678, 8889, 999 $$
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Find the Five-Number Summary The five-number summary consists of: minimum, first quartile (Q1), median, third quartile (Q3), and maximum.
- Minimum: The smallest number is $1$.
- Maximum: The largest number is $3556788999$.
- Median: The median is the middle value. There are 30 data points, so the median is the average of the 15th and 16th values. The values are $18$ and $26$, hence $$ \text{Median} = \frac{18 + 26}{2} = 22 $$
- First Quartile (Q1): The median of the first half (1 to 15). The 7th value is $5$.
- Third Quartile (Q3): The median of the second half (16 to 30). The 22nd value is $223$.
Thus, the five-number summary is: $$ \text{Minimum} = 1, \ Q1 = 5, \ \text{Median} = 22, \ Q3 = 223, \ \text{Maximum} = 3556788999 $$
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Determine the Mode The mode is the value that appears most frequently. Looking at our data, the number $5$ appears most frequently (3 times), thus: $$ \text{Mode} = 5 $$
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Calculate the Midhinge The midhinge is the average of Q1 and Q3: $$ \text{Midhinge} = \frac{Q1 + Q3}{2} = \frac{5 + 223}{2} = 114 $$
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Calculate the Mean To calculate the mean using the relationship among mode, median, and mean, we can use: $$ \text{Mean} = \frac{3 \cdot \text{Median} + \text{Mode}}{4} $$ Substituting the values: $$ \text{Mean} = \frac{3 \cdot 22 + 5}{4} = \frac{66 + 5}{4} = \frac{71}{4} = 17.75 $$
The five-number summary is:
- Minimum: $1$
- Q1: $5$
- Median: $22$
- Q3: $223$
- Maximum: $3556788999$
Mode: $5$
Midhinge: $114$
Mean: $17.75$
More Information
The five-number summary provides a quick snapshot of the distribution of the data. The mode indicates the most common value, while the midhinge gives a central tendency similar to the median. The mean calculated here shows a relationship influenced by the mode and median.
Tips
- Miscounting the values when calculating the median or quartiles.
- Not sorting the data in ascending order before finding the summary statistics.
- Confusing the formulas for mean, mode, and midhinge.
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