STATISTICS

SOME BASIC STATISTICS MEAN (AVERAGE) – Add all of the data together and divide by the number of elements within that set of data. MEDIAN – The middle number in a set of data that has been arranged from least to greatest. MODE – The number that appears the most often in a set of data. If no element appears more than once the set is set to have NO MODE. If there is a tie then you list all of the numbers that are tied as the mode.

M & M & M EXAMPLE Lets see if we can calculate the mean, median, and mode for: 2, 3, 5, 6, 7, 7, 12 First the mean. We need to find the sum of our list and then divide by the number of numbers in our list = / 7 = 6 (Mean)

2, 3, 5, 6, 7, 7, 12 Now for the median. The median is the middle of our list. Sometimes it is easy to tell by looking but if not I like to use the following calculation. Take the number of numbers in our list and add 1 to it. Then divide by 2. That tells you that the number in that position in the list is your median. For this example there are 7 numbers so I take which is 8. I then divide by 2 which gives me 4. So the 4 th number in our list is the median. Here it is 6!

M & M & M EXAMPLE 2, 3, 5, 6, 7, 7, 12 Finally the mode. The mode is the number that appears most often in the list. In our example it is 7. Remember if there is a tie you have to list all of them and if no number appears more than once we say there is No Mode.

FIVE NUMBER SUMMARY

WHAT IS IT? A five number summary is a statistical tool used to quickly summarize and gain insight about a set of data. The five number summary consists of the upper and lower extremes, the median, and the upper and lower quartiles.

SO WHAT DOES THAT MEAN? The upper and lower extremes are the biggest and smallest numbers that occur in the set. The median is the middle term when the data is arranged from least to greatest. The upper and lower quartiles are the median (middle) of the upper and lower halves of the data, respectively.

HOW DO I FIND IT? Lets try an example. Consider the set (1,3,4,5,6,7,9). Make sure your list is in numerical order from smallest to biggest!

LOWER AND UPPER EXTREMES The lower and upper extremes are simply the smallest and biggest numbers from your set. In our example the smallest number is 1 and the biggest number is 9. 1, 3, 4, 5, 6, 7, 9

MEDIAN (THE MIDDLE) The way I use to find the middle of a set of numbers is to count how many numbers are in your set. In our example there are 7 numbers. 1, 3, 4, 5, 6, 7, 9 Next I add one to that number and divide by 2. So = 8 and 8 / 2 = 4. That means that the 4 th number in our list is the middle. That would be 5! 1, 3, 4, 5, 6, 7, 9

WHERE ARE WE? So far we have found the smallest and biggest numbers in our list along with the middle. 1, 3, 4, 5, 6, 7, 9

UPPER AND LOWER QUARTILES 1, 3, 4, 5, 6, 7, 9 You will notice that the median (the middle) cuts our list into two parts. The only remaining thing to do is find the upper and lower quartiles. The lower quartile is simply the middle of the lower half and the upper quartile is the middle of the upper half.

UPPER AND LOWER EXTREMES (CONTINUED) 1, 3, 4, 5, 6, 7, 9 1, 3, 4 represents the lower half of our data. The middle is pretty easy to determine just by looking but if need be we can use the formula we used before. Count the numbers in our list (3), add 1 and divide by = 4 and 4 / 2 = 2. That means the second number in the list (3) is our lower quartile. 6, 7, 9 represents the upper half and through the same process we find the upper quartile is 7.

SUMMARY So for our example the five number summary is: –Smallest Number: 1 –Lowest Quartile (Middle of the lower half): 3 –Median (Middle): 5 –Upper Quartile (Middle of the upper half): 7 –Biggest Number: 9 A little side note for you. Many of you have worked with the box and whisker plot before. The box and whisker is a graphical display of the 5 number summary!

ONE MORE??? Find the five number summary for the set (1,3,4,6,8,9,10,14). –What is different about this example?????

The smallest and biggest numbers are easy enough but in this example the middle falls in between two numbers. –1, 3, 4, 6, 8, 9, 10, 14 –In this case we would need to take the average (mean) of the two numbers that it falls between. In this example the middle would be the average of 6 and 8 which would be 7.

1, 3, 4, 6, 8, 9, 10, 14 Now for the middle of the lower and upper halves! You can see that the lower half and the upper half both consist of 4 numbers so the middle of each half falls in between numbers again. 1, 3, 4, 6 The middle of the lower half falls in between 3 and 4 so we would need to take the average between 3 and 4 which gives us a lower quartile of , 9, 10, 14 The middle of the upper half falls between 9 and 10. Thus the upper quartile would be the average of 9 and 10 which would result in the upper quartile being 9.5. So in summary for this example –Smallest Number: 1 –Lower Quartile (Middle of Lower Half): 3.5 –Median (Middle): 7 –Upper Quartile (Middle of Upper Half): 9.5 –Biggest Number: 14