Mean,+Median,+and+Mode

**Intro:** __Mean, median, and mode are all considered "measures of central tendency." This means that they are used to describe the typical or middle value in a given set of values. __ = **Mean** =
 * Definition:** The mean is commonly referred to as the "average". It is defined as the sum of all data values, divided by the number of data values. For example, if three data values are given (90, 80, and 70), the mean will be 80 because (90+80+70)/3= 80

The mean is **82.5** - The average would be: Sum of data values: (94+86+82+81+79+73) = 495, divided by the # of data values which is 6.
 * Sample Questions:**
 * 1:** Given this data set, what would be the mean?
 * Student: || Score on Mr. Killian's Math Test ||
 * Jim || 94 ||
 * Ted || 86 ||
 * Donald || 82 ||
 * Tyrone || 81 ||
 * Hercules || 79 ||
 * Tom || 73 ||

http://stattrek.com/descriptive-statistics/central-tendency.aspx http://www.mathsteacher.com.au/year8/ch17_stat/02_mean/mean.htm#stat

∑ X*P(X) Basically you have to multiply each outcome by the probability of its occurrence, then sum them up. This will give you the mean, or "average," of the outcome. Example: Look at example 2 here.
 * Probability Distribution Mean:**

=Median=
 * Definition:** The middle number in a sequence of numbers. The manner in which the median is determined depends on the amount of values in the sequence.

The median would be **3** - The median evenly divides the lower half and the upper half of the sequence.
 * Sample Questions:**
 * 1. Given the set of numbers:** 1,2,3,4,5

The median would be **5** - **Because there is an even number of numbers in the sequence, there is not a single middle number.** - **4** and **6** are the numbers closest to the middle (**the middle divides the upper half and lower half into equal parts),** so to find the median we must find the mean of the two numbers.
 * 2. Given the set of numbers:** 1,4,6,7

The median would be **10.5**
 * 3. Given the set of numbers:** 1,4,6,10,11,15,20,35

There are 8 numbers in the sequence, therefore there is no middle number. The lower half would be: 1,4,6,10 The upper half would be: 11,15,20,35 The two middle numbers would be: 10,11 The mean of these two numbers is: 10.5
 * Why?**

http://www.purplemath.com/modules/meanmode.htm

=Mode=
 * Definition:** In a set of numbers, the **mode** is the one that occurs the most frequently.

The mode is 75. - It is the result that appears the most often.
 * Sample Questions:**
 * 1:** The following is the results from Mr. Killian’s most recent math test. {69, 75, 54, 69, 75, 75, 73, 55, 12, 35, 75}. What is the mode?

The mode is 64 **and** 43. - This question actually is **bimodial**, meaning that there are two modes. Since both 64 and 43 appear the same amount of times, they are both modes.
 * 2:** The weatherman forecasted temperatures of 55, 64, 43, 67, 43, 64, and 42 for the next week. What is the mode of this data?

There is no mode - In this problem, each number appears once. Therefore, there is no mode.
 * 3:** In the following set of numbers, what is the mode? {1,2,3,4,5,6,7,8,9}

http://www.mathgoodies.com/lessons/vol8/mode.html

=Bringing it All Together:=

1. Find the mean, median and mode of: 1, 5, 5, 6, 9

Possible Answers
 * = a. ||< Mean- 5, median- 5, mode- 5 ||
 * = b. ||< Mean- 5.2, median- 5, no mode ||
 * = c. ||< Mean- 5.2, median- 5, mode- 9 ||
 * = d. ||< Mean- 5.2, median- 5, mode- 5 ||
 * = e. ||< Mean- 5.2, no median, mode- 5 ||

2. For a class test, the mean score was 65, the median score was 71, and the standard deviation of the scores was 7. The teacher decided to add 5 points to each score due to a grading error. Which of the following statements must be true for the new scores? (A) None (B) I only (C) II only (D) I and II only (E) I, II, and III
 * I. || The new mean score is 70. ||
 * II. || The new median score is 76 ||
 * III. || The new standard deviation of the scores is 12. ||

**3. What is the median for the numbers: 34, 29, 112, 87** A. 29B. 87C. 58D. 112E. 60.5

**4. The term test scores of 15 students enrolled in a Business Statistics class were recorded in ascending order as follows: 4, 7, 7, 9, 10, 11, 13, 15, 15, 15, 17, 17, 19, 19, 20 After calculating the mean, median, and mode, an error is discovered: one of the 15\'s is really a 17. What measure of central tendency will change?:** (http://classof1.com/solution-library/view/statistics/descriptive-statistics/Multiple-choice-questions-based-on-Weighted-average-arithmetic-mean-mode-and-median/6591/Descriptive-statistics/string/tag)
 * 1) the mean only
 * 2) the mode only
 * 3) the median only
 * 4) the mean and mode
 * 5) all three measures
 * 6) banana

**Answers:****1. D****2. D****3. E****4. D**