Finding the Sample Median

Finding the Sample Median • Find the sample median x ~ Illustration A: An odd number of data • Given: The distance, in feet, run in five seconds by second graders during a fitness evaluation test was recorded as: 41, 48, 27, 55, 31, 45, 51

Ranking the Data 3rd 5th smallest largest 2nd 4th 6th 27, 31, 41, 45, 48, 51, 55 1st 2nd 3rd 4th 5th 6th 7th • Since the median is the “middle value”, the data must first be ranked in order of value • Typically, ranking is smallest value first and largest value last: (Do you have your sample data ready to use?) largest 3rd 5th smallest 2nd 4th 6th Sample data = {41, 48, 27, 55, 31, 45, 51} Ranked data = { }

The Formula • Next, the depth (position from end) of the median, d(x), is determined using the formula: ~ n+1 2 ~ d(x)= 7 n = 7 1 2 3 4 5 6 7 +1 2 8 2 ~ d(x)= 4 = = n+1 2 ~ d(x)= Ranked data = { 27, 31, 41, 45, 48, 51, 55 } 7 1 2 3 4 5 6 7 n 7 7 = 4

Determining the Median Value Fromthe smallest value data Ranked data ={ } 27, 31, 41, 45, 48, 51, 55 Fromthe largest value data • The value of the median is determined by locating the data in the 4th position of the ranked data and observing its value: Position 1 Position 2 Position 3 Position 4 Position 4 Position 3 Position 2 Position 1 • The median can also be determined by locating the data in the 4th position from the largest

The Answer! Fromthe smallest value data Position 1 Position 2 Position 3 Position 4 Ranked data ={ } 27, 31, 41, 45, 48, 51, 55 Fromthe largest value data Position 4 Position 3 Position 2 Position 1 • Notice that the same data is located from either end, which means you can find the median one way and use the other as a check The median distance is 45 feet

Finding the Sample Median • Find the sample median x ~ Illustration B: An even number of data • Given: The distance, in feet, ran in five seconds by preschoolers during a fitness evaluation test was recorded as: 6, 10, 13, 11, 12, 8, 8, 11

Ranking the Data 4th 7th Smallest 2nd 3rd 6th 5th Largest 6, 8, 8, 10, 11, 11, 12, 13 1st 2nd 3rd 4th 6th 5th 7th 8th • Since the median is the “middle value”, the data must first be ranked in order of value • Typically, ranking is smallest value first and largest value last: (Do you have your sample data ready to use?) Smallest Largest 4th 5th 7th 2nd 3rd 6th Sample data = {6, 10, 13, 11, 12, 8, 8, 11} Ranked data = { }

The Formula • Next, the depth (position from end) of the median, d(x), is determined using the formula: ~ n+1 2 ~ d(x)= 8 n = 8 1 2 3 4 5 6 7 8 +1 2 9 2 ~ d(x)= 4.5 = = n+1 2 ~ d(x)= Ranked data = { 6, 8, 8, 10, 11, 11, 12, 13} 8 1 2 3 4 5 6 7 8 n 8 8 = 4.5

Determining the Median Value ~ d(x)= 4.5 ~ • The .5 part of d(x) indicates the median value is half way between the values of the data in the4th and 5th positions of the ranked data: Fromthe smallest value data 10 11 = 10.5 + 2 21 2 ~ ~ x x = = Position 1 Position 2 Position 3 Position 4 Position 5 Ranked data = {6, 8, 8, 10, 11, 11, 12, 13} 10+11 10 11 = 10.5

The Answer! Fromthe smallest value data Position 1 Position 2 Position 3 Position 4 Position 5 Ranked data = {6, 8, 8, 10, 11, 11, 12, 13} Fromthe largest value data • As before, the median can also be determined by locating the data in the 4.5th position from the largest: Position 5 Position 4 Position 3 Position 2 Position 1 • Notice that the same two data are located from either end The median distance is 10.5 feet

Finding the Sample Median