Download Mean - Illinois State University Department of Psychology

Reasoning in Psychology
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Psychology 138
2017
• Summaries or pictures of the distribution
• Numeric descriptive statistics
– Shape: modality, and skew (and kurtosis, not cover much)
– Measures of Center: Mode, Median, Mean
– Measures of Variability (Spread): Range, Inter-Quartile Range,
Standard Deviation (& variance)
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Descriptive statistics
• Useful to summarize or describe distribution
with single numerical value.
– Value most representative of the entire
distribution, that is, of all of the individuals
– Central Tendency: 3 main measures
– Mean (M)
– Median (Mdn)
– Mode
• Note: “Average” may refer to each of these three
measures, but it usually refers to Mean.
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Measures of Center
• Most commonly used measure of center
• Arithmetic average
– Computing the mean
– Formula for population mean
(a parameter):
– Formula for sample mean
(a statistic):
–Note: Mean is mathematical
result, not necessarily score on
scale (e.g., average of 2.5
children)
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The Mean
åX
m=
N
åX
M= X =
n
Divide by the
total number in
the population
Add up all of
the X’s
Divide by the
total number in
the sample
– Conceptualizing the mean
As the center of
As the representative
the distribution
score in the distribution
Reasoning in Psychology
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The Mean
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
Reasoning in Psychology
Using Statistics
The Mean
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
Reasoning in Psychology
Using Statistics
The Mean
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
Reasoning in Psychology
Using Statistics
The Mean
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
Balancing
point
Reasoning in Psychology
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The Mean
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
1 2 3 4 5 6 7 8 9 10
1+10 = 11
Mean = 11/2 = 5.5
Reasoning in Psychology
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Balancing
point
The Mean
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
1 2 3 4 5 6 7 8 9 10
Balancing
points
Reasoning in Psychology
Using Statistics
The Mean
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
1 2 3 4 5 6 7 8 9 10
Reasoning in Psychology
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The Mean
What happens if we add an
observation to our
distribution?
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
1 2 3 4 5 6 7 8 9 10
Reasoning in Psychology
Using Statistics
The Mean
What happens if we add an
observation to our
distribution?
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
What happens if we add an
observation to our
distribution?
Reasoning in Psychology
Using Statistics
The Mean
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
What happens if we add an
observation to our
distribution?
Reasoning in Psychology
Using Statistics
The Mean
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
1+10+7 = 18
Mean = 18/3 = 5.5
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What happens if we add an
Balancing observation to our
distribution?
point
The Mean
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
1+10+7 = 18
Mean = 18/3 = 6.0
Reasoning in Psychology
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The Mean
What happens if we add an
observation to our
distribution?
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
1+10+7 = 18
Mean = 18/3 = 6.0
Reasoning in Psychology
Using Statistics
The Mean
What happens if we add an
observation to our
distribution?
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
1+10+7 = 18
Mean = 18/3 = 6.0
Reasoning in Psychology
Using Statistics
The Mean
What happens if we add an
observation to our
distribution?
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
1 2 3 4 5 6 7 8 9 10
1+10+7 = 18
Mean = 18/3 = 6.0
Reasoning in Psychology
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What happens if we add an
observation to our
New
Balancing distribution?
point
The Mean
– Conceptualizing the mean
As center of
As the representative
To be fair, let’s give
distribution
score in the distribution
everybody the
Girl Scout bake sale for camping trip
same amount.
1 2 3 4 5 6 7 8 9 10
$12
$25
$30
$6 $18
$15
12+25+30+6+18+15+13=119
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The Mean
$13
119/7 = 17
– Conceptualizing the mean
As center of
As representative score
distribution
in distribution
Girl Scout bake sale for camping trip
1 2 3 4 5 6 7 8 9 10
$17
$17
$17
$17 $17
$17
12+25+30+6+18+15+13=119
$17
119/7 = 17
So everybody is represented by same
score, the mean is the “standard”
17+17+17+17+17+17+17=119 119/7 = 17
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The Mean
• Suppose that you combine 2 groups together.
– How do you compute new group mean?
Average the 2
averages
91.6 + 77.2
X=
= 84.4
2
But it only works
this way when the
two groups have
exactly the same
number of scores
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A weighted mean
• Suppose that you combine 2 groups together.
$205!?
– How do you compute new group mean?
I only
have $191
Group 1
Group 2
New Group
X1 = 17
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X2 = 24
17 + 24
XN =
= 20.5
2
A weighted mean
• Suppose that you combine 2 groups together.
– How do you compute new group mean?
Group 1
Group 2
New Group
12+25+30+6+18+15+13+25+17+30=191
X1 = 17
X2 = 24
Mean = 191/10 = 19.1
$30
$12
$25
$6
$17
$30
$12
$30
$25
$13
$18
$18
$15
$13
$17
$15
Reasoning in Psychology
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A weighted mean
$6
$30
$25
$25
The mean
thecombine 2 groups together.
• Suppose
that is
you
score in new group mean?
–representative
How do you compute
the distribution
Group 1
Group 2
New Group
X1n1 + X 2 n 2
X1 = 17
X2 = 24
XN =
n1 + n 2
(17 * 7) + (24 * 3)
=
= 19.1
7+3
$17
$17
$17
$17
$24
$24
$24
$17
$17
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$17
A weighted mean
• Change/add/delete a given score, then the mean
will change.
– Suppose that one of the girl scouts discovered that she
had really made $23 instead of $30. So now the total is
119-7=112 112/7 = $16 (instead of $17)
5 10 15 20 25 30
17
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Characteristics of a mean
• Change/add/delete a given score, then the mean
will change.
– Suppose that one of the girl scouts discovered that she
had really made $23 instead of $30. So now the total is
119-7=112 112/7 = $16 (instead of $17)
17
Reasoning in Psychology
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Characteristics of a mean
• Change/add/delete a given score, then the mean
will change.
– Suppose that one of the girl scouts discovered that she
had really made $23 instead of $30. So now the total is
119-7=112 112/7 = $16 (instead of $17)
5 10 15 20 25 30
16
Reasoning in Psychology
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Characteristics of a mean
• Change/add/delete a given score, then the mean
will change.
• Add/subtract a constant to each score, then the
mean will change by adding(subtracting) that
constant.
– Suppose that you want to factor out a $2 camping fee
for each girl scout. Subtract 2 from each amount. Now
the total is $105, so the mean is 105/7 = $15.
– But notice you could have just subtracted $2 from the
previous mean of $17 and arrived at the same answer.
Reasoning in Psychology
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Characteristics of a mean
• Change/add/delete a given score, then the mean
will
change.
– Suppose
that the troop sponsor agreed to match the money made
by each girl scout (they give each girl scout an additional amount
• Add/subtract
a constant to each score, then the
of money equal to however much each made on the sale). So now
mean
willischange
adding(subtracting)
the total
$238, andby
the mean
for each girl is 238/7 =that
$34
– Which is 2 times the original mean
constant.
• Multiply (or divide) each score by a constant, then
the mean will change by being multiplied by that
constant.
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Characteristics of a mean
• Median divides distribution in half: 50% of
individuals in distribution have scores at or below
the median.
– Case1: Odd number of scores
Step1: put scores in order
$12
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$25
$30
$6 $18
$15
The median
$13
•
Median divides distribution in half: 50% of
individuals in distribution have scores at or
below the median.
– Case1: Odd number of scores
Step1: put scores in order
Step2: find middle score
$6
$12 $13
$15
$18 $25
$30
That’s the median,
a score on scale
Reasoning in Psychology
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The median
• Median divides distribution in half: 50% of
individuals in distribution have scores at or below
the median.
– Case2: Even number of scores
Step1: put scores in order
Step2: find middle 2 scores
$6 $12
$13
$15
$18
$18 $25
15 + 18
= 16.5
2
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The median
$30
Step3: find arithmetic
average of 2 middle scores
That’s the median
Note: mathematical result not a
score on scale
• Mode: score or category with greatest frequency.
– Pick variable in frequency table or graph with highest
frequency (mode always a score on scale).
Mode = 5
Modes = 2, 8
T-shirt size
30
Mode = Medium
25
20
15
10
5
0
Small
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The mode
Medium
Large
X-large
• Depends on a number of
factors, like scale of
measurement and
shape.
– The mean is the most
preferred measure and it is
closely related to
measures of variability
– However, there are times
when the mean is not the
appropriate measure.
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Which center when?
• If data on nominal scale: Mode only
–
–
–
–
Unranked categories (e.g. eye color)
Not a numeric scale
Can not do arithmetic operations on values
Can not calculate cumulative percentages
Eye color
Mode = Brown
Median =
Green
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Which center when?
• If data on ordinal scale: Median (plus Mode)
– Not a numeric scale (e.g., T-shirt size)
– Can not do arithmetic operations on values
– Can calculate cumulative percentages on frequencies
(median is score at 50th percentile)
Median of T-shirt size = Medium
Mode of T-shirt size = Medium
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Which center when?
• If data on interval or ratio scale BUT:
– Distributions open-ended
• Response category like “5 or more”
• Extreme values unknown, so can not calculate mean
– Distributions skewed with long tails
• Extreme values over influence mean
• E.g., income sample of 50
– 47 middle income ($60,000-$100,000) and 3
millionaires or billionaires
– Median = $80,000
– Mean = $135,000 or $60,000,000
• Median (plus Mode)
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Which center when?
• If data on interval or ratio scale AND no
exclusionary conditions: Mean (plus Median) (plus Mode)
– Numeric scale
– Can do arithmetic calculations on values
– Have benefit of other statistics using the mean, such as
standard deviation
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Which center when?
Situation
Most
Representative
Nominal
Mode
Ordinal
Median
Mode
Skewed interval or
ratio
Median
Mode
Open ended interval
or ratio
Median
Mode
Interval or Ratio
Mean
Mode
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In Summation
Least
Representative
Can’t Use
Median/Mean
Mean
Mean
• Impact of shape on center (interval or ratio scale)
mean = median = mode
Positively skewed distribution
mean > median > mode
mean = median, 2 modes
Negatively skewed distribution
mean < median < mode
Mean & median pulled toward tail
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Which center when?
Mode
Median
Mean
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0-10,000
45,734
?
175-200,000
261,600
325,212
Chicago distributions
Check out your hometown: http://www.city-data.com/
price of
Mode The average
0-10,000
houses in this
Medianneighborhood
45,734 is …
Mean
?
buying
175-200,000
261,600
325,212
selling
When you say “average” are you talking
about the median or the mean?
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Buyer beware: Know your distribution
• Today’s lab
– Compute mean, median, & mode both by hand
& using SPSS
• Questions?
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Wrap up