Probability & Statistics Box Plots

Describing Distributions Numerically Five Number Summary and Box Plots (Box & Whisker Plots )

Box Plots Median - the center The median is the middle value that divides the histogram into two equal areas. To find the median:  Put the data in order from least to greatest  If there are an odd number of entries, the median is the middle number.  If there are an even number of entries, the median is the mean of the two middle numbers.

Box Plots Example: Find the median of each set of data: a) 14.1, 3.2, 25.3, 2.8, -17.5, 13.9, 45.8 First, put the data in order! -17.5, 2.8, 3.2, 13.9, 14.1, 25.3, 45.8 b) 6, 8, 9, 9, 4, 8, 5, 9, 2, 5, 3, 4 2, 3, 4, 4, 5, 5, 6, 8, 8, 9, 9, 9 First, put the data in order! The median is the middle number The median is the mean of the two middle numbers. The median is 13.9 The median is 5.5

Try this. Find the median of each set of data: (show work) b)18, 22, 20, 25, 24, 24 a) 6, 8, 10, 9, 7

Box Plots Range - the spread Identify the maximum and minimum values of your data. The range is the minimum value subtracted from the maximum value. Range = Max - Min

Box Plots Quartiles The median splits your data into two halves. (A lower half and an upper half.) The first (lower) quartile, Q1 is the median of the lower set of data and the third (upper) quartile, Q3 is the median of the upper set of data , 2.8, 3.2, 13.9, 14.1, 25.3, 45.8 median Q1 Q3 The median is not included in the upper or lower half of data. lower half upper half

Box Plots Interquartile Range (IQR) A measure of Spread! The interquartile range (IQR) is the difference between the first and third quartile. IQR = Q3 - Q , 2.8, 3.2, 13.9, 14.1, 25.3, 45.8 Q1 Q3 IQR = 25.3 – 2.8 IQR = 22.5 For the previous set of data:

Box Plots median 5.5 The median is not included in the upper or lower half of data. Q1 = 4 Q3 = 8.5 IQR = IQR = 4.5 In this case, the median was not even part of the data. 2, 3, 4, 4, 5, 5, 6, 8, 8, 9, 9, 9 2, 3, 4, 4, 5, 5, 6, 8, 8, 9, 9, 9 Ex.)Find the IQR of the following data.

Box Plots Five Number Summary The five number summary of a set of data includes the following:  Minimum  First (Lower) Quartile  Median  Third (Upper) Quartile  Maximum

Box Plots Example: Find the five number summary for the following set of data. 22, 6, 2, 5, 4, 8, 15, 10, 15, 20 2, 4, 5, 6, 8, 10, 15, 15, 20, 22 Minimum = Q1 = Median = Q3 = Maximum = 2 22 The minimum and maximum values are the easiest to find. Remember to put the data in order.

Box Plots Example: Find the five number summary for the following set of data. 22, 6, 2, 5, 4, 8, 15, 10, 15, 20 2, 4, 5, 6, 8, 10, 15, 15, 20, 22 Minimum = Q1 = Median = Q3 = Maximum = 2 22 Next, we need to find the median. 9 With two numbers in the middle, we need to find the number halfway between them. (Find the mean of the two numbers.)

Box Plots Example: Find the five number summary for the following set of data. 22, 6, 2, 5, 4, 8, 15, 10, 15, 20 Minimum = Q1 = Median = Q3 = Maximum = 2 22 Finally, we can find the quartiles. 9 2, 4, 5, 6, 8, 10, 15, 15, 20, 22 Q1 Q3 2, 4, 5, 6, 8,10, 15, 15, 20,

Box Plots Example: Find the five number summary for the following set of data. 22, 6, 2, 5, 4, 8, 15, 10, 15, 20 Minimum = 2 Q1 = 5 Median = 9 Q3 = 15 Maximum = 22 2, 4, 5, 6, 8,10, 15, 15, 20, 22 This is the five number summary for this data.

Box Plots Creating a Box Plot The five number summary, contains almost all of the information we need to create a box plot (or box & whisker plot). Additionally, we will need the IQR in order to identify any outliers in our data.

Box Plots Creating a Box Plot If the data does not have any outliers, the five number summary contains all the information needed. The box plot is drawn above a number line. (Number lines can also be drawn vertically.)

Box Plots Creating a Box Plot If the data does not have any outliers, the five number summary contains all the information needed. Start by drawing line segments perpendicular to the number line at the location of the median and the quartiles. Q1 Q3 Median

Box Plots Creating a Box Plot If the data does not have any outliers, the five number summary contains all the information needed. Median Q3 Q1 Use horizontal lines to connect the top and bottom of the line segments forming the “box”.

Box Plots Creating a Box Plot If the data does not have any outliers, the five number summary contains all the information needed. Median Q3 Q1 The “whiskers” are drawn using the maximum and minimum values. (As long as they are not outliers.) Draw a dot at the location of the maximum and minimum value, then connect the dot to the box using a line segment. Maximum (within range) Minimum (within range) Not an outlier

Box Plots Five Number Summary Minimum = 2 Q1 = 5 Median = 9 Q3 = 15 Maximum = 22 2, 4, 5, 6, 8,10, 15, 15, 20, 22 Example: Draw a box plot to represent the following data. If you recall, we already found the five number summary for this data. The number line must range from 2 to 22; a range of 20. With 10 tick marks available, we can do this by using a scale of Median = 9Q1 = 5Q3 = 15Min = 2Max = 22

Try this. Find the five number summary for the following set of data. Then, draw a box plot to describe the data. 6, 18, 20, 25, 15, 15, 15, 12, 12, 22, 9, 10, 18 Minimum = Q1 = Median = Q3 = Maximum =

Box Plots If a box plot has outliers, they are noted as open circles and are not used for the maximum and minimum of the graph. In the next lesson, we will learn to set up “fences”. Any data that falls outside of the fences, is an outlier.

Box Plots The following is what a box plot would look like if it had one outlier. Outlier Fences The fences are drawn exactly 1.5 IQRs away from the box. The distance between Q1 and Q3 is 1 IQR. 1 IQR 1.5 IQRs ½ IQR Q1Q3

Box Plots The following is what a box plot would look like if it had one outlier. Median Q1Q3 Minimum (Within range) Maximum (Within range) Outlier Lower Fence Upper Fence This value was the highest number in the data set. Because it lies outside the fences, it is an outlier. This value was the second highest number in the data set. Because it is the highest number within the fences, it is used as the maximum for the box plot.