Mean+Median+Mode

=__Mean Median Mode__ = When analyzing a set a numbers, it is easier to use those numbers by determining one number that will represent the entire set. One way of finding one representative number that tends to show the middle of a set of numbers is by finding the central tendency to the set. The three most common forms of central tendency are mean, median, and mode.

Use this set of numbers in the examples… 2, 3, 3, 3, 5, 6, 8, 8, 10, 12

**__Mean __** - The mean or average of a set of //n// numbers is found by adding all of the numbers in the set and dividing by //n//. Ex. 2+3+3+5+6+8+8+10+12= 60 you can also use the formula where the the numerator is the sum 60/10= 6 of all the numbers and n is the number of the numbers in the set 6 is the mean of the set Ex. Since there are 10 numbers in the set, which is an even number, the median is between 5 and 6. In this case the median is 4.
 * __Median __** - The median of a set of //n// numbers is the number that is in the middle of the set when the numbers are put in numerical order, either increasing or decreasing. If //n// is an even number then the median is the average of the two numbers in the middle.

**__Mode __** -    The mode of a set of //n// numbers is the number that appears most often. If each number only shows up once than there is no mode. If, for example, there are two numbers that both appear the most then they are both modes of the set. When a set of numbers has more than one mode, the set is called multimodal. Ex. The mode of this set of numbers is 3 because it appears 3 times which is more than any other number. If another 8 was added to set, which would mean that there were three 3’s and three 8’s than there would be 2 modes, 3 and 8. It is important to understand when to use each of the central tendencies to accurately illustrate each specific set of numbers. While there are some cases in which mean will work well in representing a set of numbers, there are also times when the median or the mode with represent a set of numbers better; or more closely to the actual set of numbers. 50 is obviously an outlier and distant from the other values -by finding the mean you would end up with 11 (which is greater than all of the values except one showing that it is not a good use for a central tendency) -by finding the median you would end up with 5 (which is a much more reasonable number to represent the central tendency of this set of data ** Graphically Shown: **  When data is given symmetrically, the mean, median, and mode are all equal. Graphically they all meet at the same point.  When data is given in a skewed fashion, with outliers for example, the graph is then altered. The median is unwavering, and will always be found in the middle of the data, making it accurate in all circumstances, but more time consuming when the sets of data become larger. The mean on the other hand can be affected greatly when introduced with an outlier in the data, as shown on the graph. For a negatively skewed set of data, mean<median<mode. And, for a positively skewed set of data, mode<median<mean. Sometimes the exact values of data are not given and you are only given grouped measures. In this case they will usually give you a percentage and then two values that that percentage can be found between. You can still find the mean of this grouped data.
 * Central Tendency || When to use them… ||
 * Mean || * When data is symmetrically distributed ||
 * Median || * When data is skewed
 * When sets of data contain ouliers*
 * This is mostly seen with information on home prices and incomes ||
 * Mode || * When dealing with categorical data (surveys) ||
 * outliers- are numbers are extreme values that would cause the mean to be less accurate than the median.
 * For example… with the set of being: 2, 3, 4, 5, 6, 7, 50
 *  Finding the Mean and Median from grouped data using a frequency table: **
 * ** Class Interval ** || ** Frequency (f) ** || ** Midpoint (x) ** || ** fx ** ||
 * ** $1-5 ** || 6  ||  2  ||  12  ||
 * ** $6-10 ** || 5  ||  8  ||  40  ||
 * ** $11-15 ** || 4  ||  20  ||  80  ||
 * ** $16-20 ** || 3  ||  12  ||  36  ||
 * ** $21-25 ** || 4  ||  6  ||  20  ||
 * || ** n =22=

** ||  || ** Σ fx = = 188 ** || x = midpoint of interval fx = sum of measurements of interval Σ fx = sum of all the measurements of the intervals Use the formula: and substitute the information

188/22= $8.54 In order to find the median of a grouped set of data it is more of an approximation. This is because the actual values may not be known. When approximating the median there is a different formula. Use the formula: L= lower class limit of the interval that contains the median n= total # of measurements w= class width fmed= frequency of class with median Σfb= sum of frequencies of classes before the class with the median



__Here are some sample questions that you might see on the SATs:__ 1. Consider a data set of 10 positive values where there is only one value that appears twice and the rest appear once. Which of the following central tendencies will be changed when each value in this data set is multiplied by a constant whose absolute value is greater than I. Mean II. Median III. Mode a. I only b. II only c. III only d. I, II, and III e. II and III

Use this table of grades scored on a math test to answer the question.

Student A 88 Student B 95 Student C 53 Student D 88 Student E 93

2. What if Student F took the test late. If Student F got a score of 88, which of the central tendencies would change? I. Mean II. Median III. Mode a. I only b. II only c. III only d. I, II, and III e. II and III

Answers:

1 : e 2 : a

Sources: http://www.quickmba.com/stats/centralten/ http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/Measures-of-Central-Tendency.topicArticleId-25951,articleId-25905.html http://mathworld.wolfram.com/Mean.html


 * Using 5-Number Summaries, Box Plots, or Histograms are helpful ways to organize and visualize Mean, Median, and Mode.