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Chapter 10—Data, Statistics, and Probability
102 Statistics
This lesson is currently under development.
Topics Covered in this Lesson  Introduction to Statistics
 Average Values
 Median Values
 Mode Values

Average Values
Definition The average, or arithmetic mean, of a group of numbers is the center point of all those number values.  Notes  Average and arithmetic mean are simply two different terms for the same thing.
 Arithmetic mean is pronounced as arithMETik, and not as arITHmetik.
 Arithmetic mean is often spoken more simply as the mean.
  
Procedure To find the average, or arithmetic mean, of a set of numbers:  Add the given values
 Divide the sum by the number of values.

Examples of Calculating Averages
Examples and Exercises
Determine the average, or mean, value of the set of numbers shown here.  Show all you work on a sheet of paper.
 Continue the exercises until you can work them without making mistakes.
 
Data is not always presented in purely numerical form.
Example
 Consider the bar graph shown here. You can see five units labeled A through G. Their value range from 2 to 6. What is the average value? A  2  B  3  C  5  D  6  E  6  Sum  22   Average = 22/5 = 4.4  
Example
 What is the mean temperature as indicated on this graph? 60  50  70  70  70  80  50  40  50  60   Points = 10 Sum = 660 Average = 660/10 66°C  
The statistical mean (or average) is supposed to provide some insight into ....
If the average annual temperature in a certain city is 48°F, you have a pretty good idea what the temperatures are like  even if the local temperature might drop below 0°F a couple of nights in the winter and above 110°F for a few days in the summer. The average value tells you nothing about extreme temperatures  only that all the cold days, hot days, and everything in between averages out to 48°F.
One of the major shortfalls of the arithmetic mean is that a single, really crazy outofbounds value can have a significant impact on
Example
In order for a group of fives students to qualify for an important scholastic competition, their group average on a qualifying exam must be at least 85%. If four of those students have scores of 75%, 72%, 80%, and 85%. What must the fifth student score in order to achieve the necessary group average.
Referring to the equation for arithmetic mean:
 = 85 The desired group averages
 X_{1}, x_{2}, x_{3}, x_{4} = 75, 72, 80, 85 The values that are known
 n = 5 The total number of grades in the group
 x_{5} = The unknown value; the minimum grade the fifth student must attain.
Substituting these values into the equation for the mean:
85 =  75 + 72 + 80 + 85 + x_{5} 
5 
Solving for the needed grade, x_{5}
85(5) – (75 + 72 + 80 + 85) = x_{5}
x_{5 }= 425  312 =
Median Values
Definition The median value is the exact middle value of a set of numbers. 
Procedure To find the median value of a set of numbers: When there is an odd number of values,  Arrange the numbers in numerical order.
 Find the value in the middle of the list.
That value is the median value  Odd number of values  When there is an even number of values,  Arrange the numbers in numerical order.
 Locate the two middle numbers in the list.
 Find the average of those two middle values.
That is the median value  Even number of value  
Example
Examples and Exercises
Determine the median value for data having an odd number of values.  Show all you work on a sheet of paper.
 Continue the exercises until you can work them without making mistakes.
 
Example
Examples and Exercises
Determine the median value for data having an even number of values. Round the result to the nearest 10th (1 decimal place),  Show all you work on a sheet of paper.
 Continue the exercises until you can work them without making mistakes.
 
Example
Consider the bar graph shown here. You can see five units labeled A through G. Their value range from 2 to 6. What is the median value? The bars are already arranged in order of increasing values. It should be clear that item C is the middle value  5  
Example
 What is the median temperature as indicated on this graph? The data, as presented on the graph, does not show the temperatures in sequence. So it is necessary to build a table that does show the values in sequence:  40   50   50   50   60   60   70   70   70   80    Then locate the values in the exact middle of the list. The value is 60. The median temperature is 60°C 
The Mode
Definition The mode value is the value that occurs the largest number of times. 
Examples
1. Determine the mode for this set of data:
1,2,12,12,3,4,5,67
The value 12 occurs more often than any other. So it is the mode.
2. Determine the mode for this set of data:
1,2,8,12,3,4,5,6
There is no mode value.
Important: When there are no repeated values, there simply is no mode. THE MODE IS NOT ZERO! See the next example. 
3.Determine the mode for this set of data:
0,1,2,3,0,4,5,6,0,7,9,9,0
The value 0 occurs four times, so the mode is zero.
4. Determine the mode for the following set of data:
0,1,1,2,3,1,4,3,3,2,6
There are two mode values  1 and 3
 The four examples shown above demonstrate the following facts about the mode:
 The mode is the value (or values) that occurs most often in a set of data.
 When all values occur the same number of times, there is no mode value.
 When there are two or more values that occur the same number of times, each is a mode.
Examples and Exercises
Consider the bar graph shown here. You can see five units labeled A through G. Their value range from 2 to 6. What is the mode? Visual inspection shows that items D and E have identical values. There are no identical units, so the median value is 6.  
Example
 What is the mode of the temperatures shown on this graph? Arranging the values in sequence helps locate multiple instances of the same values by visual inspection.  40   50   50   50   60   60   70   70   70   80    You can see that 50 and 70 occur three times, 60 occurs two times, and the other values only once. So there are two modes: 50°C and 70°C. 