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Chapter 10—Data, Statistics, and Probability

10-2 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 ar-ith-MET-ik, and not as ar-ITH-me-tik.
  • Arithmetic mean is often spoken more simply as the mean.

 

Procedure

To find the average, or arithmetic mean, of a set of numbers:

  1. Add the given values
  2. 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
 

66C

 


The statistical mean (or average) is supposed to provide some insight into ....

If the average annual temperature in a certain city is 48F, you have a pretty good idea what the temperatures are  like -- even if the local temperature might drop below 0F a couple of nights in the winter and above  110F 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 48F.

One of the major shortfalls of the arithmetic mean is that a single, really crazy out-of-bounds 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
X1, x2, x3, x4 =  75, 72, 80, 85  The values that are known
n = 5  The total number of grades in the  group
x5 = 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 + x5
5

Solving for the needed grade, x5

85(5) (75 + 72 + 80 + 85) = x5

x5 = 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,
  1. Arrange the numbers in numerical order.
  2. 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,
  1. Arrange the numbers in numerical order.
  2. Locate the two middle numbers in the list.
  3. 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 60C

 

 

 

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:  50C and 70C.

 

 

David L. Heiserman, Editor

Copyright   SweetHaven Publishing Services
All Rights Reserved

Revised: June 06, 2015