Download Mean - NMSU College of Business

Data Transformation
• Data conversion
• Changing the original form of the data to a
new format
• More appropriate data analysis
• New variables
Data Transformation
Summative Score =
VAR1 + VAR2 + VAR 3
Descriptive Analysis
• The transformation of raw data into a form
that will make them easy to understand
and interpret; rearranging, ordering, and
manipulating data to generate descriptive
information
Tabulation
• Tabulation - Orderly arrangement of data
in a table or other summary format
• Frequency table
• Percentages
Frequency Table
• The arrangement of statistical data in a
row-and-column format that exhibits the
count of responses or observations for
each category assigned to a variable
Central Tendency
Type of Scale
Nominal
Ordinal
Interval or ratio
deviation
Measure of
Central
Tendency
Measure of
Dispersion
Mode
Median
Mean
None
Percentile
Standard
Base
• The number of respondents or
observations (in a row or column) used as
a basis for computing percentages
Index Numbers
• Score or observation recalibrated to
indicate how it relates to a base number
• CPI - Consumer Price Index
Measures of Central Tendency
• Mean - arithmetic average
– µ, Population;
, sample
X
• Median - midpoint
of the distribution
• Mode - the value that occurs most often
Population Mean
Xi

N
Sample Mean
 Xi
X
n
Measures of Dispersion
or Spread
•
•
•
•
Range
Mean absolute deviation
Variance
Standard deviation
The Range
as a Measure of Spread
• The range is the distance between the
smallest and the largest value in the set.
• Range = largest value – smallest value
Deviation Scores
• The differences between each observation
value and the mean:
d x x
i 
i 
Low Dispersion Verses High
Dispersion
5
Low Dispersion
4
3
2
1
150
160
170 180
190
Value on Variable
200
210
Low Dispersion Verses High
Dispersion
5
4
High dispersion
3
2
1
150
160
170
180
190
Value on Variable
200
210
Average Deviation
(X i  X )
0
n
Mean Squared Deviation
 ( Xi  X )
n
2
The Variance
Population

2
Sample
S
2
Variance
 X  X )
S 
n 1
2
2
Variance
• The variance is given in squared units
• The standard deviation is the square root
of variance:
Sample Standard Deviation
S
2
  Xi X 

n 1
The Normal Distribution
• Normal curve
• Bell shaped
• Almost all of its values are within plus or
minus 3 standard deviations
• I.Q. is an example
Normal Distribution
13.59%
2.14%
34.13%
34.13%
13.59%
2.14%
Normal Curve: IQ Example
70
85
100
115
145
Standardized Normal Distribution
• Symetrical about its mean
• Mean identifies highest point
• Infinite number of cases - a continuous
distribution
• Area under curve has a probability density = 1.0
• Mean of zero, standard deviation of 1
Standard Normal Curve
• The curve is bell-shaped or symmetrical
• About 68% of the observations will fall
within 1 standard deviation of the mean
• About 95% of the observations will fall
within approximately 2 (1.96) standard
deviations of the mean
• Almost all of the observations will fall
within 3 standard deviations of the mean
A Standardized Normal Curve
-2
-1
0
1
2
z
The Standardized Normal is the
Distribution of Z
–z
+z
Standardized Scores
z
x

Standardized Values
• Used to compare an individual value to the
population mean in units of the standard
deviation
z
x

Linear Transformation of Any Normal
Variable into a Standardized Normal
Variable




Sometimes the
scale is stretched
X
Sometimes the
scale is shrunk
z
-2
-1
0
1
2
x
