Download Measure of Center

Measure of Center
A measure of center is a value at the center or
middle of the data set
Surprising, huh?
Mean
Arithmetic mean (or just mean)
A measure of center found by adding the data
values and dividing by the number of values
This is what is usually called an average.
That term is imprecise, and we should stick
with mean
Notation

Sigma - means “add these up” (sum)
x
The variable used to represent the data values
n
The size of the sample (number of data values)
N
The size of the population (number of data values)
x

x
n
x


N
Mean of the sample data
Read: x-bar
Mean of the population values (all of them)
Read: Like “you” with an “m”
Example
6 students are asked how long they studied
last week.
Data: 7, 8, 10, 11, 13, 25
x 7  8  10  11  13  25

x

 12.3
n
6
So for our sample, the mean hours studied
was 12.3 hours
Note
The mean can be strongly influenced by
outliers, since it takes into account every
value.
Data: 7, 8, 10, 11, 13, 25
Mean: 12.3
Median
The median is the middle value when the data
is listed in order. Median is denoted ~
x
If the number of data values is odd, the
median is the middle data value
If the number of data value is even, there is no
middle data value, so we find the mean of
the two numbers in the middle
Example
Data: 5, 6, 8, 11, 13, 25
Even number of data values, so the median is found by finding
the mean of the two middle numbers, 8 and 11. (8+11)/2 = 9.5
Data: 5, 6, 8, 11, 13, 25, 26
Odd number of data values, so the median is the middle value, 11
Mode
The mode of the data is the value that occurs most
often. The mode is denoted by M
It is possible to have one mode, two modes
(bimodal), many modes (multimodal), or no
modes at all (when no data is repeated)
The mode is most commonly used with data at the
nominal level, since it is the only measure of
center that can be done.
Midrange
Midrange is a mostly useless measure of
center. It is determined by finding the mean
of the highest and lowest data values.
highest  lowest
midrange 
2
Rule for rounding
Never round while doing your calculations
(more than absolutely necessary)
Round your final answer so that it has one
more decimal place than the original data
did.
This is primarily so our mean doesn’t imply
we know more than we actually do.
Example
For the data: 5, 6, 8, 11, 13, 25
we found the mean to be 11.3
The exact value was 11.33333333333333333
If we rounded to 11.33, it would suggest that
we had measured minutes, not just hours.
We don’t want to make our mean appear
more accurate than it is.
Skewness
and the measures of center
Our Coffee Data
Hours Slept by Caffeine Drinkers
14
12
Frequency
Mean: 6.38 hrs
Median: 6.85 hrs
Modes: 5.3, 7.2, 8.2
10
8
6
4
2
0
1
3
5
7
Hours Slept
Distribution: Strongly skewed to the left
9
11
Homework
2.4: 3, 9
Think about: 22, 23, 24