**Learning Objectives**

After completing this tutorial, you should be able to:

- Find the mean of a list of values.
- Find the median of a list of values.
- Find the mode of a list of values.
- Find the range of a list of values.
- Find the standard deviation of a list of values.
- Use a frequency distribution to find the mean.

** Introduction**

In this tutorial we will be looking at basic concepts of central tendencies. We will go over how to find the mean, median and mode of a list of values as well as the range and standard deviation. I think you are ready to get started on these central tendencies.

** Tutorial**

You can find the mean by adding up all the values and then dividing that sum by the number of values that you have.

There is only one mean to a list of values.

The mean may or may not be a number that is in the original list of values.

This does not necessarily mean it is the middle number in the original list. You need to make sure that your values are in numeric order from smallest to largest before you find the median.

There is only one median to a list of values.

The median may or may not be a number that is in the original list of values.

You can have more than one mode, if more than one value occurs the same amount of times and that is the highest occurrence.

Find the mean, median and mode of the quizzes.

So we need to sum up all of the quizzes and then divide by 8, since there are 8 quizzes:

***Add numerator**

***Divide by 8**

'**The mean is 85.125.**

**We need to list the numbers in numeric order:**

**75, 75, 79, 80, 88, 93, 95, 96**

If we pick 80 for our median we have 3 values below it and 4 above it. If we pick 88 for our median then we have 4 values below it and 3 above it. So neither of those values are the median. This does not mean we don’t have a median.

Note how there is an even number of values listed. If that is the case, we need to draw a line down the middle of the list and take the mean of the two numbers next to that line:

**75, 75, 79, 80 | 88, 93,
95, 96**

The mean of 80 and 88 is

It helps to list the numbers in order to find the mode.

**75, 75,
79, 80, 88, 93, 95, 96**

Note how 75 occurs two times, which is the value that occurs the most.

**75 is the mode.**

Find the mean, median and mode of the points.

So we need to sum up all of the points and then divide by 5, since there are 5 games:

***Add numerator**

***Divide by 5**

'**The mean is 5.4.**

**We need to list the numbers in numeric order:**

**3, 3, 6, 6, 9**

This time we have an odd number of values. Our median is going to be 6 (the first 6 listed). That number has two values above it and two below it, so it is the middle value.

**6 is the median. ** It is the value that is
right smack dab
in the middle of this list of values.

It helps to list the numbers in order to find the mode.

**3, 3, 6, 6,
9**

Note how both 3 and 6 occur two times, which is the most.

**Both 3 and 6 are the mode.**

This time we are given the mean and we need to find one of our values.

Keep in mind that this is still a mean problem. We
will still
use the idea that we need to sum up the exams and then divide it by 5
to
get the mean. We can let our unknown exam be *x*.

***Solve for x (missing
test)**

***Inverse of div. by 5 is mult.
by 5**

***Inverse of add 385 is sub. 385**

This is similar to example 3, except that the test
score counts twice
instead of one time. So when we set this up we need to make sure
that we notate that properly.

***Inverse of div. by 5 is mult.
by 5**

***Inverse of add 254 is sub. 254**

One way to measure dispersion (variability) among
numerical values
is to find the range of those numbers. **The range of a set of
numerical
data points is the difference between the largest value and the
smallest
value.** In other words you take the greatest measurement minus
the least measurement.

Another way to measure dispersion of a data set is to
find the standard
deviation of its values. **The standard deviation is a relative
measure of the dispersion of a set of data. **

Note that the range only involves two values in its calculation - the highest and the lowest. However, the standard deviation involves every value of its data set.

The steps to finding the standard deviation are as
follows:

**Step 1: Find the mean of the values
of the data set.**

**Step 2: Find the difference between
the mean and each
separate value of the data set.**

**Step 3: Square each difference found
in step 2.**

**Step 4: Add up all of the squared
values found in step
3.**

**Step 5: Divide the sum found in
step 4 by the
number of data values in the set.**

**Step 6: Find the nonnegative square
root of the quotient
found in step 5.**

I don't know about you, but I find it easier to work
with a group of
numbers like this when they are in chronological order. Let's put
them in order from lowest to highest: 3, 3, 4, 4, 8, 8, 8, 8, 10,
12, 20.

**Let's find the range.**
What
do you think it is?

Looking at the difference between the largest value,
which is 20 and
the smallest value, which is 3, it looks like **the range is 17. **

Now lets tackle the **standard
deviation**.

So we need to sum up all of the values and then divide
by 11, since
there are 11 numbers:

***Add numerator**

***Divide by 11**

Sometimes there are a lot of values in a data set and
some of them
are repeated. In that case, it may be easier to group those
values
using a frequency distribution. **This is a chart that lists
each
unique value and then next to the number indicates the frequency, or
number
of times, that value occurs in the data set.**

For example, if you had the list of test scores for a
class:

75, 80, 90, 80, 75, 75, 50, 65, 65, 50, 100, 90, 100, 90, 75, 40, 60,
60

**Writing these values ( x)
in a frequency
(f) distribution chart you would have:**

This combines two ideas covered in this tutorial,
finding the **mean** and setting up a **frequency distribution**.

As shown above, the **frequency distribution** for
this set of numbers
is

As requested, I’m going to use the frequency
distribution to set up
my mean formula. Instead repeating numbers in my sum, I’m going
to
indicate a repetition by taking that value times the number of times it
occurs in the list. For example, 75 occurs 4 times. Instead
of writing it out 4 times in my sum, I will find 75(4) which is the
equivalent.

***calculate numerator**

***divide**

** Practice Problems**

These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. **Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.**
In fact there is no such thing as too much practice.

To get the most out of these, **you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem**. At the link you will find the answer
as well as any steps that went into finding that answer.

Practice Problem 1a: Find the mean, median, and mode.

1a. The number of cd’s sold by Dave’s Discs for the last 6 days are given in the table.

Find the mean, median, and mode.

(answer/discussion
to 1a)

Practice Problem 2a:Find the test score.

2a. A student received scores of 92, 83, and 71
on three quizzes.
If tests count twice as much as quizzes, what is the lowest score that
the student can get on the next test to achieve a mean of at least 80?

(answer/discussion
to 2a)

Practice Problem 3a:Find the range and standard deviation of the list of scores that were made by a football team during a season.

3a. 7, 21, 21, 17, 17, 14, 7, 0

(answer/discussion to 3a)

(answer/discussion to 3a)

Practice Problem 4a:Find the mean of the frequency distribution.

** Need Extra Help on these Topics?**

**Go to Get
Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for
some
more suggestions.**

Last revised on August 7, 2011 by Kim Seward.

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