NOTES TO CONSIDER BEFORE ATTEMPTING EX 2C BOX PLOTS

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1 NOTES TO CONSIDER BEFORE ATTEMPTING EX 2C BOX PLOTS A box plot is a pictorial representation of the data and can be used to get a good idea and a clear picture about the distribution of the data. It shows how the data is set out. To draw a box plot you need the values of: the median, the two quartiles (lower and upper) and the two extremes (highest and lowest). This is called the five figure summary or five number summary. Whisker Interquartile range Whisker Lowest value (minimum) Lower quartile Median Upper quartile Highest value (maximum) Range The number line is drawn so that it covers all the values in the set of data The rectangular 'box' represents the interquartile range (the middle half of the data) The 'box' is drawn from the lower quartile (Q L or Q 1 ) to the upper quartile (Q U or Q 3 ) The vertical line drawn in the 'box' represents the median (middle value) A whisker extends from each end of the 'box' to the lowest and highest values Example 1 Number line (with a suitable scale) This data shows Tom's cricket scores over the last 13 games. Show the data below as a box plot. (We are finding the box plot without outliers. Outliers will be treated later.) Step 1: Find the median Sort the data from lowest to highest Middle value is the 7th value There are 13 scores The middle score = the median = 46 Step 2: Find the upper and lower quartiles. Divide the data in half. If the data set have even number of values, it can be divided equally. However, in this question, there are 13 values in the data set 13 (odd number) scores don't divide the data set evenly into two groups, so we leave the middle score, 46 out.

2 There are 6 scores in the lower half. The median of these scores =!"!!" = 39! We call this the lower quartile Q L or Q 1 There are 6 scores in the upper half. The median of these scores =!"!!" = 55! We call this the lower quartile Q U or Q 3 Five number summary Lowest score (minimum) = 20 runs Lower quartile = 39 runs Median = 46 runs Upper quartile = 55 runs Highest score = 90 runs Use the results to draw a box plot The box plot shows how the scores are spread out. A boxplot divides the data into four groups, each with about the same number of data scores The range of the box is called the interquartile range (IQR) o IQR = Q 3 -Q 1 = = 16 runs o The middle half of the scores fall in this range. o Eliminates the extreme values in the top and bottom of the data o Helps to measure the consistency of the data 25% of scores are less than Q 1 or 75% of scores are above Q 1 50% of scores are less than the median or 50% of scores are above the median 75% of scores are less than Q 3 or 25% of scores are above Q 3 Don't mix up between the range and the interquartile range (IQR). The range is the difference between the Highest and Lowest values. The interquartile range (IQR) is the difference between the Upper and Lower quartiles. Finding the Measures of Centre for Grouped Data

3 Interpreting Information from a Box Plot Example 2 This boxplot represents test results fro Year 8 Science. The test was marked out of 40. a. Read off the approximate values and interpret them. b. What can you say about the test results? c. Comment on the skewness. a. & b. Statistic Approximate Interpretation value Median 9 It is the middle value of the date 50% of students received a result above 9 and 50% of students received a result below 9 Upper Quartile 17 one quarter or 25% of the students received a result greater than 17 Lower Quartile 6 one quarter or 25% of the students received a result less than 6 Range 35 highest score lowest score = 35-0 = 35 the results ranged from 0 to 35 Interquartile 11 IQR = upper quartile lower quartile = 17 6 = 11 Range 50% of the students received a result between 6 and 17 the middle half of the data has a spread of 11 Comment: We can see that 75% of the class got a score below 17. Not a vey pleasing result. Perhaps the test was too hard or too long or the students did not prepare properly. d. The data is positively skewed.

4 Outliers / Limits / Fences A box plot can help us find outliers. Outliers are data values which are found outside the main group of the data. The lines outside the box are called whiskers. If the whiskers are more than 1.5 times the width of the box, then there are outliers in the sample. The first step is to identify the Lower and Upper Limits / Fences by using the following formula: Lower limit / fence = Q 1 (1.5 IQR) Upper limit / fence = Q 3 + (1.5 IQR) Outliers are therefore data values that are: less than the lower limit or upper limit / fence The whiskers are extended to the last actual data values before the outliers. Lower limit IQR 1.5 IQR 1.5 IQR Upper limit Outlier Outliers TESTING FOR OUTLIERS Step 1: Establish the values fo Q 1, Q 3 and IQR. Step 2: Calculate Q IQR. This value is called the lower limit or fence. Step 3: Check to see whether there are any data values less than the lower limit / fence. If so, then each value is an outlier. Outlier < Lower limit value Step 4: Calculate Q IQR. This value is called the upper limit or fence. Step 5: Check to see whether there are any data values greater than the upper limit / fence. If so, then each value is an outlier. Outlier > Upper limit value Note: To receive full marks for a box plot question, you must clearly show all relevant information including IQR, Lower and Upper limit calculations.

5 Example 3 Using the data from Example 1 (Tom s cricket scores over the last 13 games) Lowest score, minimum = 20 runs Lower quartile, Q 1 = 39 runs Median, Q 2 = 46 runs Upper quartile, Q 3 = 55 runs Highest score, maximum = 90 runs Remember: To get full marks, you must show all relevant values e.g. IQR, limit calculations, etc. Step 1: Q 1 =39 and Q 3 = 55 IQR = = 16 Step 2: Lower limit / fence: Q x IQR = x 16 = = 15 Step 3: Check the data. There are no data values less than 15, and so there are no outliers on the left side of the box plot.the whisker will extend to the last data value on this side. Thus it extends to the value 20. Step 4: Upper limit fence: Q x IQR = x 16 = = 79 Step 5: Check the data. There are two data values more than 79. These values are 82 and 90 and so these values are outliers on the right side of the box plot. They will be represented by dots or asterisks. The whisker will extend to the last data value before 82. Thus the whisker extends to 56 (third highest value). Min Q 1 Med Q 3 Max Each outlier is represented by an asterisk *

6 Box Plots for grouped data in frequency tables (GROUPED data) Example 6 Draw a box plot for the goals scored in 100 games as shown in the table. Find the range, interquartile range, and comment on the spread of the data. Goals in a game Total Frequency Cumulative To find the median and the upper and lower quartiles, set up a Goals Frequency frequency cumulative frequency column We have 100 values which divides in half evenly, 50 in each = 20 half. There is no middle value so the median is the average of the 50 th and the 51 st values The 50 th value = 2 goals, the 51 st value = 3 goals Thus the median = = 2!3 5 = 2. 5 goals The lower quartile is the median of the lower 50 values which is the average of the 25 th and 26 th values. The 25 th and 26 th values both equal 2 goals, so Q 1 = 2 goals The upper quartile is the median of the upper 50 values which Total 100 is the average of the 75 th and 76 th values. The 75 th and 76 th values both equal 4 goals, so Q 3 = 4 goals. The box plot shows the range (Max - Min) = 6-0 = 6 goals The interquartile range is: Q 3 Q 1 = 4-2 = 2 goals The upper half of the data is slightly more spread out than the lower half Exercise 2C

7 NOTES TO CONSIDER BEFORE ATTEMPTING EX 2D & 2E Relating A Box Plot To Distribution Shape A box plot gives an idea of the skewness of a distribution. Always look at the box first ie. between Q 1 and Q 3 to see shape of distribution. Whiskers are used to confirm shape. NB: If whiskers contradict shape, use box to decide on shape. Approximately symmetric distribution The box plot is symmetric The median is generally in the middle of the box The whiskers are approximately equal in length If the mean, median and the mode are equal then it is said to be symmetric about the mean. Mode = Median = Mean Positively skewed distribution The box plot has the median off-centre and generally to the left. The left-hand whisker will be short and the right-hand whisker will be long. When the long tail is to the right, the mean is pulled away from the main body of he data set to the right; and it is located to the right of the median. In this case, the data set is positively skewed because of the predominance of the data values towards the positive end of the distribution. Mode < Median < Mean Negatively skewed distribution The box plot has the median off-centre and generally to the right. The right-hand whisker will be short and the left-hand whisker will be long. When the long tail is to the left, the mean is pulled away from the main body of the data set to the left, and it is located to the left of the median. In this case, the data set is negatively skewed because of the predominance of the data values towards the negative end of the distribution. Mean < Median < Mode

8 Comparing boxplots Comparing Box Plots - It is very important to be able to COMPARE all the key features when comparing box plots. Here are the math results for four year 7 classes. To be able to compare they must be PARALLEL box plots, so that they all use the same scale. Centre - Median Class 7B has the highest median mark of approximately 78. 7A has the lowest median mark of approximately 62. Shape - Skew You can describe skew by looking at just the middle 50% or the entire range of data. To be safe make sure you are clear what you have compared: Class 7D is approx. symmetrical and 7B is positively skewed with the median score being a little closer to the lower quartile and also the minimum score. Class 7C could be seen as negatively skewed if we look at just the box (middle 50%) but the upper whisker is longer than the lower. Class 7A has its median equally spaced between the upper and lower quartile scores but the upper whisker is much longer than the lower, meaning there is more spread of data above the median rather than below. So class 7A it is positively skewed. Maximum and Minimum 7B has the highest mark of around 98. All other classes have equal highest marks of 90 7D and 7A share lowest marks of 40. Overall Range Class 7B has and 7C have approximately the same lowest range. Class 7A and 7D have the highest range. Upper and Lower Quartiles The 7B upper quartile is equal to the maximum of the other classes. Interquartile Range Class 7D has the highest IQR. By comparing the middle 50% all other IQR's are similar. In summary Class 7B is the highest performing class because it has 25% of its students achieving higher than any grade of the other classes. Half of its students achieve higher than three quarters of Class 7A and 7C. The lower quartile of 7B is above the median for all other classes, with the bottom 25% of marks being higher than the lower quartile for all other classes. Exercise 2D & 2E