Five Number Summary and Box Plots

Five Number Summary and Box Plot Basics The Five Number Summary consists of the minimum, lower quartile (Q1), median, upper quartile (Q3), and maximum. It is used to determine the variability (or the differences in data) of a data set and to construct box plots. Box plots are used as a visual representation of the data.

Five Number Summary Definitions Minimum: the smallest value in a data set Lower quartile (Q1): the 25th percentile; 25% of the information is less than this value Median: the 50th percentile or middle value Upper quartile (Q3): the 75th percentile; 75% of the information is less than this value Maximum: the largest value in a data set

Example First, you’ll break up this information into quarters. 5 6 7 8 9 10 11 15 18 19 20 24 First, you’ll break up this information into quarters. There will be the same amount of numbers in each section.

Example Find the middle number 5 6 7 8 9 10 11 15 18 19 20 24 Find the middle number Since this list has 13 values, the middle number would be the 7th number. Note: If there was an even amount of values, you would find the average of the two middle numbers.

Example 5 6 7 8 9 10 11 15 18 19 20 24 Now that you have your halfway point, find the middle of the top and bottom sections. There are six values in the top section so the middle value would fall between the 3rd and 4th values. Find the middle value for the bottom section.

Example 5 6 7 8 9 10 11 15 18 19 20 24 There are 6 values in the bottom section. The middle value would fall between the 10th and 11th values. Now that we have our intervals set up, we can find the values for our five number summary.

Example Middle of the top: 7.5 Middle of the bottom: 18.5 Aka Q1 5 6 7 8 9 10 11 15 18 19 20 24 Middle of the top: 7.5 Aka Q1 Middle of the bottom: 18.5 Aka Q3

Example 5 6 7 8 9 10 11 15 18 19 20 24 The minimum is 5. To find the lower quartile (Q1), find the average of 7 and 8. Q1 is 7.5 The median, or middle value, is 10. To find the upper quartile (Q3), find the average of 18 and 19. Q3 is 18.5. The maximum is 24

Example 5 6 7 8 9 10 11 15 18 19 20 24 You would write the five number summary for this data set as follows: Minimum = 5 Q1 = 7.5 Median = 10 Q3 = 18.5 Maximum = 24

Making a Box Plot Minimum = 5 Q1 = 7.5 Median = 10 Q3 = 18.5 Maximum = 24 Using the five number summary, you can easily construct a box plot. First, we need to make a number line. Choose your minimum, maximum, and scale based on your five number summary Our minimum is 5 and maximum is 24. What should we choose as our minimum, maximum, and scale for our number line?

Making a Box Plot Minimum = 5 Q1 = 7.5 Median = 10 Q3 = 18.5 Maximum = 24 0 would be a good choice for our minimum, 25 for our maximum with a scale of 5. Next, we need to mark off the values for each number in the five number summary. 0 5 10 15 20 25

Making a Box Plot Make a small dot for the minimum and maximum. Q1 = 7.5 Median = 10 Q3 = 18.5 Maximum = 24 Make a small dot for the minimum and maximum. Make tick marks for Q1, median, and Q3. Connect the tick marks with two lines to form a box Connect the dots with lines to the box in the center 0 5 10 15 20 25

Describing a Box Plot Minimum = 5 Q1 = 7.5 Median = 10 Q3 = 18.5 Maximum = 24 In a box plot, each segment represents 25% of the information. What can you tell about the way the information is grouped based on this graph? 25% 25% 25% 25% 0 5 10 15 20 25

Describing a Box Plot Minimum = 5 Q1 = 7.5 Median = 10 Q3 = 18.5 Maximum = 24 Intervals that are smaller (like from the minimum to Q1) have information that is tightly packed together. Intervals that are larger (like from the median to Q3) have information that is more spread out. 25% 25% 25% 25% 0 5 10 15 20 25

Describing a Box Plot Minimum = 5 Q1 = 7.5 Median = 10 Q3 = 18.5 Maximum = 24 The length of the box (from Q1 to Q3) represents the IQR, or interquartile range. This is the middle 50% of the data 50% 0 5 10 15 20 25

Describing a Box Plot Minimum = 5 Q1 = 7.5 Median = 10 Q3 = 18.5 Maximum = 24 This value will help you determine the variability of a data set or to compare variability of more than one set of data. 50% 0 5 10 15 20 25

Describing a Box Plot Minimum = 5 Q1 = 7.5 Median = 10 Q3 = 18.5 Maximum = 24 The larger the IQR, the larger the variability of the data set. The smaller the IQR, the smaller the variability. The IQR for this box plot is 11 (Q3-Q1). 18.5 – 7.5 = 11 50% 0 5 10 15 20 25

Describing a Box Plot Minimum = 5 Q1 = 7.5 Median = 10 Q3 = 18.5 Maximum = 24 You can also look at the length of the box to help determine the variability of the data. 50% 0 5 10 15 20 25

About what percentage of students scored between 70 and 90 on the test depicted in the box plot above? 40 50 75 90 Cannot be determined from the information given

Describing a Box Plot Minimum = 5 Q1 = 7.5 Median = 10 Q3 = 18.5 Maximum = 24 Along with the five number summary, you can also talk about whether the box plot is skewed or symmetric based on the size of each interval. Do you think this box plot is skewed right, left, or symmetric? 0 5 10 15 20 25

Describing a Box Plot Minimum = 5 Q1 = 7.5 Median = 10 Q3 = 18.5 Maximum = 24 This box plot is skewed to the right because the intervals between the median and Q3 and the interval between Q3 and the maximum are very spread out. 0 5 10 15 20 25

Describing a Box Plot Here is the correct description of this box plot Minimum = 5 Q1 = 7.5 Median = 10 Q3 = 18.5 Maximum = 24 IQR = 11 Range = 19 Skewed to the right Here is the correct description of this box plot 0 5 10 15 20 25

Complete the following problem and turn in to the teacher The following are scores from 20 students on a unit test in mathematics. Find the 5 number summary Find the interquartile range (IQR). Make a box plot for this information. Describe the distribution. 75 92 62 78 85 77 93 65 80 90 50 70 57 98 45 54 73 84