The Statistical Environment Essay

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Week 5 some notes on measures of central tendency

Mean, Median and Mode |

We use statistics such as the mean, median and mode to obtain information about a population from our sample set of observed values.
MeanThe mean (or average) of a set of data values is the sum of all of the data values divided by the number of data values. That is:
Example 1The marks of seven students in a mathematics test with a maximum possible mark of 20 are given below: 15 13 18 16 14 17 12Find the mean of this set of data values.Solution:So, the mean mark is 15.
Symbolically, we can set out the solution as follows:So, the mean mark is 15.
MedianThe median of a set of data values is the middle value of the data set when it has been arranged in ascending order. That is, from the smallest value to the highest value. The Median is the 'middle value' in your list. When the totals of the list are odd, the median is the middle entry in the list after sorting the list into increasing order. When the totals of the list are even, the median is equal to the sum of the two middle (after sorting the list into increasing order) numbers divided by two. Thus, remember to line up your values, the middle number is the median! Be sure to remember the odd and even rule.
Example 2The marks of nine students in a geography test that had a maximum possible mark of 50 are given below: 47 35 37 32 38 39 36 34 35Find the median of this set of data values.Solution:Arrange the data values in order from the lowest value to the highest value: 32 34 35 35 36 37 38 39 47The fifth data value, 36, is the middle value in this arrangement.
Note:
In general:If the number of values in the data set is even, then the median is the average of the two middle values.
Example 3Find the median of the following data set: 12 18 16 21 10 13 17 19Solution:Arrange the data values in order from the lowest value to the highest value: 10 12 13 16 17 18 19 21The number of values in the data set is 8, which is even. So, the median is the average of the two middle values.
Alternative way:There are 8 values in the data set.The fourth and fifth scores, 16 and 17, are in the middle. That is, there is no one middle value.
Note: * Half of the values in the data set lie below the median and half lie above the median. * The median is the most commonly quoted figure used to measure property prices. The use of the median avoids the problem of the mean property price which is affected by a few expensive properties that are not representative of the general property market.
ModeThe mode of a set of data values is the value(s) that occurs most often. The Mode is the Most Reoccurring Number in a Set.The mode in a list of numbers refers to the list of numbers that occur most frequently. A trick to remember this one is to remember that mode starts with the same first two letters that most does. Most frequently - Mode. You'll never forget that one!The mode has applications in printing. For example, it is important to print more of the most popular books; because printing different books in equal numbers would cause a shortage of some books and an oversupply of others.Likewise, the mode has applications in manufacturing. For example, it is important to manufacture more of the most popular shoes; because manufacturing different shoes in equal numbers would cause a shortage of some shoes and an oversupply of others.
Example 4Find the mode of the following data set: 48 44 48 45 42 49 48Solution:The mode is 48 since it occurs most often.To find the mode from a set of numbers, I have 5, 9, 7, 7, 6, 5, 2, 3, and 5, arrange then in order from smallest to largest and find the number that occurs more than the most in the sequence, in my case 5 is the mode. If you have a set of numbers that does…