Mean, Median, Mode and Range

Measures of Central Tendency Measure of central tendency – A value that represents the centre of a data set, can be the mean, median or mode.   Data Set – A collection of numbers Ex: [9,13,12,8,8] is a data set

Mean The “mean” is computed by adding all of the numbers in the data together and dividing by the number of elements contained in the data set. Example: Data Set = 2, 5, 9, 3, 5, 4, 7 Number of Elements in Data Set = Mean =

Median The “median” of the data set depends on whether the number of elements in the data set is odd or even. Steps to find the median: Reorder the data set from smallest to largest. If the number of elements is odd the median is the middle number. If the number of elements is even the median is the average of the two middle numbers.

Median Example #1: Example #2: Find the median of the data set: 9,13,12,8,8   Written in increasing order = 8, 8, 9, 12, 13 Median = 9 Example #2: Find the median of the data set: 6,2,7,19,13, 9   Written in increasing order = 2, 6, 7, 9, 13, 19 Median = (7+9) ÷ 2 = 8

Mode The mode for a data set is the element that occurs the most often. It is not uncommon for a data set to have more than one mode. Some sets have no mode. Example #1: [9,13,12,8,8] Mode = 8 Example #2: [6,2,7,19,13, 9] Mode = no mode   Example #3: [2,2,6,8,7,9,8] Mode = 2 and 8

Range The range for a data set is the difference between the largest value and the smallest value of a data set. Example #1: [9,13,12,8,8] 13 – 8 = 5 Example #2: [6,2,7,19,13, 9] 19 – 2 = 17   Example #3: [2,2,6,8,7,9,8] 9 – 2 = 7

Common Errors Not arranging the data in ascending order prior to determining the median. The median in an even-numbered data set is not both middle numbers, but the average of the two. When no mode exists, the mode is not 0.

Which measure of central tendency should I use? Advantage Disadvantage Mean Commonly used Easy to calculate Useful when comparing Skewed by extreme values Median Extreme values do not affect the median Difficult to arrange large sets of data in order Mode Extreme values do not affect the mode Useful when data values are limited. Unless there is only one mode the results can be difficult to make sense of.

Example The weekly salaries of six employees at McDonald’s are $140, $220, $90, $180, $140, $200. For these six salaries find the mean, median and mode. List the data in order: 90, 140, 140, 180, 200, 220 Mean: 90+ 140+ 140+ 180 + 200 + 220 = $161.67 Median: 90,140,140,180,200,220 The two numbers that fall in the middle need to be averaged. (140 + 180) / 2 = 160 Mode: The number that appears the most is 140

Example Andy has grades of 84, 65 and 76 on three math tests. What would his score be on the next test if his average is exactly 80 on the four tests? Andy will need a 95% on his next test. (4) (80) = 225 + x 320 = 225 + x 320 – 225 = x 95 = x

Example Test scores for a class of twenty students are 93, 84, 97, 98, 100, 78, 86, 100, 85, 92, 72, 55, 91, 90, 75, 94, 83, 60, 81, 95. Find the mean, median and mode of the data set. Mean: 93+84+97+98+100+78+86+100+85+92+72+55+91+90+75+94+83+60+81+95 20 = 1709 = 85.5

Example Test scores for a class of twenty students are 93, 84, 97, 98, 100, 78, 86, 100, 85, 92, 72, 55, 91, 90, 75, 94, 83, 60, 81, 95. Find the mean, median and mode of the data set. Median: Order data: 55, 60, 72, 75, 78, 81, 83, 84, 85, 86, 90, 91, 92, 93, 94, 95, 97, 98, 100, 100 Find the average of the two middle numbers. 86 + 90 2 = 176 = 88

Example Test scores for a class of twenty students are 93, 84, 97, 98, 100, 78, 86, 100, 85, 92, 72, 55, 91, 90, 75, 94, 83, 60, 81, 95. Find the mean, median, mode, and range of the data set. Mode: 55, 60, 72, 75, 78, 81, 83, 84, 85, 86, 90, 91, 92, 93, 94, 95, 97, 98, 100, 100 The mode is 100. Range: Largest value = 100 Smallest value = 55 Range = Largest value – Smallest value Range = 100 – 55 Range = 45

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Example A store owner keep a tally of the sizes of suits purchased in her store. Which measure of central tendency should the storeowner use to describe the average size suit sold? Mean A tally was made of the number of times each color of crayon was used by a kindergarten class. Which measure of central tendency should the teacher use to determine which color is the favorite color of her class? Mode

Example A tally was made of the number of times each color of crayon was used by a kindergarten class. Which measure of central tendency should the teacher use to determine which color is the favorite color of her class? The favorite color would be the color appearing the most number of times. Use the mode.

Example You are ordering bowling shoes for a bowling alley. Which measure of central tendency would most helpful in this situation? Mode You want to know if you read more or fewer books per month than most people in your class. Which measure of central tendency would most helpful in this situation? Median You want to know the “average” amount spent per week on junk food in your class. Which measure of central tendency would most helpful in this situation? Mean

Portfolio Work Mode pg 68-69 #1,2(B),3,4,5,6 Median pg 71-72 1(A,C)2,3(B,C), Mean pg 76-77 #1,3,4 Mini-task