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Central Tendency Objectives: …to calculate measures of central tendency and to decide which measure is most appropriate for a given situation. …to construct, compare, and identify parts of a box-and-whisker plot …to construct, compare, and identify parts of a stem-and-leaf plot Assessment Anchor: 7.E.2.1 – Describe, compare, and/or contrast data using measures of mean, median, mode or range. Vocabulary alert!! CENTRAL TENDENCY – an attempt to find the “average” or “central value” of a given set of data MEAN – the sum of the data items divided by the number of data items MEDIAN – the middle data item found after sorting the data items in ascending order (could be the mean of two middle numbers if the data set has an even number of items) MODE – the data item that occurs most often RANGE – the difference between the highest and the lowest data item OUTLIER – a data item that is much higher or much lower than all the other data items Central Tendency NOTES In most cases, the data set is a list of numbers. The “average” is a single number used to identify the typical value of the data set. While the “mean” is the most common measure to use, there may be situations where either the “median” or the “mode” is a more appropriate measure of the central tendency of the data. While we want to know HOW to calculate these measures of central tendency, we would like to also focus on WHY different situations call for the use of different measures of central tendency. Sometimes the determining factor lies in the kind of data, other times it has to do with the “spread” of the data, and other times it simply comes down to what kind of argument one wishes to make using the data. EXAMPLES The following table shows the point totals for the Exeter boys’ basketball team: Game Points 1 54 2 42 3 42 4 53 5 98 6 48 7 49 8 34 9 51 10 48 11 42 MEAN: 51 <-------- add the numbers to get a total of 561, then divide by 11 MEDIAN: 48 <-------- order the numbers, then find the middle number 34, 42, 42, 42, 48, 48, 49, 51, 53, 54, 98 MODE: 42 <-------- find the number that happens most often RANGE: 64 <-------- subtract the smallest number from the largest number OUTLIER: 98 <-------- this number is MUCH higher than the rest! “So… what is the average number of points scored by the Exeter boys’ basketball team? “What if the outlier was removed?” Central Tendency The following table shows the temperatures (in ºF) for the last 10 days in March: Tues. 45º Wed. 38º Thurs. 41º Fri. 48º Sat. 46º Sun. 48º Mon. 45º Tues. 39º Wed. 36º Thurs. 40º MEAN: 42.6 <-------- add the numbers to get a total of 426, then divide by 10 MEDIAN: 43 <-------- order the numbers from smallest to largest, find the middle two numbers, add them up and divide by 2 36, 38, 39, 40, 41, 45, 45, 46, 48, 48 MODE: 45 and 48 <-------- find the numbers that happen the most! RANGE: 12 <-------- subtract the smallest from the largest number! OUTLIER: none <-------- there are no numbers MUCH higher or MUCH lower than the rest! The following table shows the test scores for students in Mr. Seidel’s period 9 class: Student 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Test Score 44 79 81 88 79 85 86 90 90 77 85 90 84 78 MEAN: MEDIAN: MODE: RANGE: OUTLIER: Central Tendency STEM-AND-LEAF PLOTS ***A stem-and-leaf plot is ONE way to organize data by showing each item in order from smallest to largest. The leaf is the last digit to the right… the stem is the remaining digit or digits. 39 3 stem 24.2 9 24 leaf stem 2 leaf To make a stem and leaf plot: GIVEN DATA: 1. List the stems and draw a line to the right of the stems. (keep stems sequential) 2. Arrange the leaves on each stem from smallest to largest. (each data item gets a leaf) 3. Include a KEY to explain how to read the plot. 38, 49, 38, 41, 50, 33, 45, 36, 53 STEM-AND-LEAF PLOT: 3 3688 4 159 5 03 Key: 4 | 5 means 45 Make a stem-and-leaf plot HR per season by Terra Coveroff: Interest rates given by 6 local banks: 46, 49, 30, 61, 39, 38, 34, 40, 33, 38 0.4%, 1.1%, 1.3%, 0.7%, 0.2%, 0.3% Central Tendency Read a stem-and-leaf plot 8 9 10 11 47 788 025559 2 1. What is the median? 2. What is the range? 3. What is the mode? Key: 9 | 7 means 9.7 27 2 4 5 5 5 28 9 9 29 0 0 2 4 5 7 1. What is the price range? 2. How many prices are $2.90 or higher? Key: 25 | 8 means $2.58 3. What is the median price? BOX-AND-WHISKER PLOTS ***A box-and-whisker plot is ANOTHER way to organize data. The data is divided into four equal parts called quartiles. The median separates the data into two halves. The lower quartile is the median of the lower half…the upper quartile is the median of the upper half. To make a box-and-whisker plot: GIVEN DATA: 1. Find the 5 SPECIAL POINTS… the lowest, the highest, the median, the lower quartile, the upper quartile 2. Draw a number line that can hold all the data. 52, 62, 90, 50, 58, 61, 35, 48, 60 35, 48, 50, 52, 58, 60, 61, 62, 90 lowest 3. Plot the special points and create the picture. • • 49 median lower quartile • 61.5 highest upper quartile • • | | | | | | | | | | | | 35 40 45 50 55 60 65 70 75 80 85 90 Central Tendency Make a box-and-whisker plot Make a box-and-whisker plot for the following numbers: 78, 90, 93, 100, 76, 88, 80, 90, 52, 78, 80, 94 Order them: 5 special points: Number line and plot: | | | | | | | | | | | | 45 50 55 60 65 70 75 80 85 90 95 100 Read a box-and-whisker plot • | | | | | 85 | | | | | 90 • || • • • | |||||| | | | 95 | | | | 100 105 110 1. What is the median? | | 2. What is the lower quartile? 3. What is the range? 4. How many data items are there? The box-and-whisker plot below shows the prices of CDs in a store: • • | 6 | | | • | 8 | | | | 10 | | | • | 12 | | | | 14 | | | • | 16 1. What is the lowest priced CD? | | 2. What is the range of the costs? 3. What is the median cost? 4. How many CDs are under $10?