Download Measures of Central Tendency & Dispersion

Measures of Central
Tendency
Section 2.3
Central Values
There are 4 values that are
considered measures of the center.
 1. Mean
 2. Median
 3. Mode
 4. Midrange

Arrays


Mean – the arithmetic average with
which you are the most familiar.
Mean:
sum of all x
x  bar 
number of x
x
x
n
Sample and Population
Symbols

As we progress in this course there
will be different symbols that
represent the same thing. The only
difference is that one comes from a
sample and one comes from a
population.
Symbols for Mean

Sample Mean:

Population Mean:
x

Rounding Rule

Round answers to one decimal place
more than the number of decimal
places in the original data.

Example: 2, 3, 4, 5, 6, 8
A Sample answer would be 4.1
Example

Find the mean of the array.

4, 3, 8, 9, 1, 7, 12
 x 4  3  8  9  1  7  12 44
x

  6.29  6.3
n
7
7
Example 2 – Use Calculator
Find the mean of the array.
2.0, 4.9, 6.5, 2.1, 5.1, 3.2, 16.6
Use your lists on the calculator and
follow the steps.
Stat, Edit – input list
Stat, Calc, One-Var Stats,
L1
Or…..(I like this way better!)



2nd Stat
Math
3: Mean (L#)
Rounding

The mean (x-bar) is 5.77.

We used 2 decimal places because
our original data had 1 decimal place.
Median
Median – the middle number in an
ordered set of numbers. Divides the
data into two equal parts.
 Odd # in set: falls exactly on the
middle number.
 Even # in set: falls in between the two
middle values in the set; find the
average of the two middle values.

Example

Find the median.

A. 2, 3, 4, 7, 8

B. 6, 7, 8, 9, 9, 10
median = (8+9)/2 = 8.5.
- the median is 4.
Ex 2 – Use Calculator

Input data into L1.
Run “Stat, Calc, One-Variable Stats,
L1”
 Cursor all the way down to find “med”

Or…….
2nd
 Stat
 Math
 4: Median(L#)

Mode

The number that occurs most often.

Suggestion: Sort the numbers in L1 to make
it easier to see the grouping of the numbers.

You can have a single number for the mode,
no mode, or more than one number.
Example
Find the mode.
 1, 2, 1, 2, 2, 2, 1, 3, 3
 Put numbers in L1 and sort to see the
groupings easier.

The mode is 2.
Ex 2

Find the mode.

A. 0, 1, 2, 3, 4

B. 4, 4, 6, 7, 8, 9, 6, 9 - 4 ,6, and 9
-
no mode
Midrange

The number exactly midway
between the lowest value and
highest value of the data set. It is
found by averaging the low and high
numbers.
Low value  High Value
midrange 
2
Example

Find the midrange of the set.

3, 3, 5, 6, 8
(3  8) 11
midrange 
  5.5
2
2
Measures of
Dispersion…..Arrays
Section 2.4
Dispersion

The measure of the spread or
variability

No Variability – No Dispersion
Measures of Variation

There are 3 values used to measure
the amount of dispersion or variation.
(The spread of the group)
1. Range
2. Variance
3. Standard Deviation
Why is it Important?

You want to choose the best brand of
paint for your house. You are
interested in how long the paint lasts
before it fades and you must repaint.
The choices are narrowed down to 2
different paints. The results are
shown below. Which paint would you
choose?

The chart indicates
the number of
months a paint
lasts before fading.
Paint A
Paint B
10
35
60
45
50
30
30
35
40
40
20
25
210
210
Does the Average Help?

Paint A: Avg = 210/6 = 35 months

Paint B: Avg = 210/6 = 35 months

They both last 35 months before
fading. No help in deciding which to
buy.
Consider the Spread

Paint A: Spread = 60 – 10 = 50 months

Paint B: Spread = 45 – 25 = 20 months

Paint B has a smaller variance which
means that it performs more consistently.
Choose paint B.
Range

The range is the difference between
the lowest value in the set and the
highest value in the set.

Range = High # - Low #
Example

Find the range of the data set.

40, 30, 15, 2, 100, 37, 24, 99

Range = 100 – 2 = 98
Deviation from the Mean

A deviation from the mean, x – x bar, is the
difference between the value of x and the
mean x bar.
We base our formulas for variance and
standard deviation on the amount that they
deviate from the mean.

We’ll use a shortcut formula – not in book.
Variance (Array)

Variance Formula
( x)
x 
2
n
s 
n 1
2
2
Standard Deviation

The standard deviation is the square
root of the variance.
s
s
2
Example – By Hand

Find the variance.
6, 3, 8, 5, 3
x
x
6
3
8
5
3
 x  25
2
36
9
64
25
9
 x
2
 143
s2 
x
2
2
( x) 2

n
n 1
25
143 
143  125 18
2
5
s 

  4.5
4
4
4
Find the standard deviation

The standard deviation is the square
root of the variance.
s  4.5  2.12
Same Example – Use
Calculator

Put numbers in L1.

Run “Stat, Calc, One-Variable Stats,
L1” and read the numbers.
Remember you have to square the
standard deviation to get variance.
Or….
2nd Stat
 Math
 7:stdDev(L1)
 Enter

Variance – By Hand

Square the ENTIRE number for the
standard deviation not the rounded
version you gave for your answer.
s  (2.121320344)  4.5
2
2
Variance on Calculator
2nd Stat
 Math
 8: Variance (L1)
