Question

Find the five-number summary and the IQR for these data: $$ 19,12,16,0,14,9,6,1,12,13,10,19,7,5,8 $$

Short answer

Answer: The five-number summary is 0, 6, 10, 14, 19, and the Interquartile Range (IQR) is 8.

Step-by-step solution

Arrange the data in ascending order

Arrange the given data in ascending order: $$ 0,1,5,6,7,8,9,10,12,12,13,14,16,19,19 $$

Find the median (Q2)

The median (Q2) is the middle value of our data set when arranged in ascending order. Our data set has 15 values, so the median is the 8th value (since (15+1)/2 = 8): $$ Q_2 = 10 $$

Find the first quartile (Q1)

The first quartile (Q1) is the median of the lower half of the data. The lower half of our data set (excluding Q2) has 7 values: \(0,1,5,6,7,8,9\) so the median of the lower half (Q1) is the 4th value (since (7+1)/2 = 4): $$ Q_1 = 6 $$

Find the third quartile (Q3)

The third quartile (Q3) is the median of the upper half of the data. The upper half of our data set (excluding Q2) has 7 values: \(12,12,13,14,16,19,19\) so the median of the upper half (Q3) is the 4th value (since (7+1)/2 = 4): $$ Q_3 = 14 $$

Determine the minimum and maximum values

The minimum and maximum values are the first and last values in the sorted data set: $$ \text{Minimum} = 0 \\ \text{Maximum} = 19 $$ The five-number summary is, therefore: 0, 6, 10, 14, 19.

Calculate the Interquartile Range (IQR)

The IQR is the difference between Q3 and Q1: $$ \text{IQR} = Q_3 - Q_1 = 14 - 6 = 8 $$ The Interquartile Range (IQR) is 8.

Key concepts

Five-Number Summary

The five-number summary is a fundamental concept in descriptive statistics that provides a quick overview of a data set. It includes five specific values that are chosen from the data and gives us valuable insights into the distribution and spread of the data. The five numbers are listed below:

• Minimum: The smallest data value. In our example dataset, the minimum is 0.
• First Quartile (Q1): The median of the lower half of the dataset. In our exercise, Q1 is 6.
• Median (Q2): The middle value when the data is ordered. For our data, the median is 10.
• Third Quartile (Q3): The median of the upper half. In this context, Q3 is 14.
• Maximum: The largest data value. Here, the maximum is 19.

Using the five-number summary helps to quickly identify the center, spread, and extreme values of the dataset. This is particularly useful in making comparisons between different datasets.

Interquartile Range (IQR)

The Interquartile Range (IQR) is a measure of statistical dispersion and is particularly useful for identifying the range within which the middle 50% of our data falls. It's calculated as the difference between the third quartile (Q3) and the first quartile (Q1).
In the example given, the IQR is computed as follows:
\[ IQR = Q_3 - Q_1 = 14 - 6 = 8 \]This calculation tells us that the spread of the middle 50% of the data spans across 8 units. The IQR is helpful as it gives a reasonable measure of the spread that is unaffected by outliers or extremely high or low values. In contrast to the full range, which considers all data points, the IQR focuses only on the central portion of the data.

Quartiles

Quartiles are values that divide a dataset into four equal parts when ordered. They are crucial in descriptive statistics for understanding the distribution of data. The role of quartiles includes:

• First Quartile (Q1): It represents the 25th percentile, meaning 25% of the data is below this point. In our example, Q1 is calculated as the median of the lower half of the data, which is 6.
• Second Quartile (Q2): This is the median of the entire dataset or the 50th percentile, indicating that half the data lies below this value. Here, Q2 is 10.
• Third Quartile (Q3): It indicates the 75th percentile, where 75% of the data values are below it. In this exercise, Q3 is 14.

Quartiles offer insight into the data's spread and can help identify potential outliers. They are vital parameters for calculating the interquartile range, which further assists in data analysis by summarizing data variability and central tendency.