Download Welcome to EDP 557 Educational Statistics

Science of Statistics
Descriptive Statistics
– methods of summarizing or describing
a set of data
tables, graphs, numerical summaries
Inferential Statistics
– methods of making inference about a
population based on the information in
a sample
Variables
Individuals are the objects described by a set of
data; may be people, animals or things
Variable is any characteristic of an individual
Statistical Data
What purpose do the data
have?
Individuals – Describe?
How many?
Variables – How many?
Definition?
Unit of
measurement?
Types of Variables
Categorical variable places an individual into
one of several groups or categories
Quantitative variable takes numerical values for
which arithmetic operations make sense
Distribution of a variable tells us what values it
takes and how often it takes these values
Exploratory Data Analysis
Examine each variable by itself… then
relationships among the variables
Start with graphs… then add numerical
summaries of specific aspects of the data
Levels of Measurement
Nominal
Ordinal
Interval
Ratio
It's important to recognize that there is a
hierarchy implied in the level of measurement
idea. At each level up the hierarchy, the current
level includes all of the qualities of the one below
it and adds something new. In general, it is
desirable to have a higher level of measurement.
In nominal measurement the numerical values
just "name" the attribute uniquely. No ordering of
the cases is implied.
For example, jersey numbers in basketball
are measures at the nominal level. Is a
player with number 30 more of anything than
a player with number 15?
In ordinal measurement the attributes can be rankordered. Here, distances between attributes do not
have any meaning.
For example, on a survey you might code
Educational Attainment as 0=less than H.S.;
1=some H.S.; 2=H.S. degree; 3=some college;
4=college degree; 5=post college. In this
measure, higher numbers mean more education.
But is distance from 0 to 1 same as 3 to 4?
In interval measurement the
attributes does have meaning.
distance
between
For example, when we measure temperature (in
Fahrenheit), the distance from 30-40 is same as
distance from 70-80. The interval between values is
interpretable. Because of this, it makes sense to
compute an average of an interval variable, where it
doesn't make sense to do so for ordinal scales. Do
ratios make sense at this level? For example, is it
twice as hot at 80 degrees as it is at 40 degrees?
Finally, in ratio measurement there is always an
absolute zero that is meaningful. This means that
you can construct a meaningful ratio.
Weight is a ratio variable. In applied social
research most "count" variables are ratio. Is
number of clients in past six months ratio?
Why?
Describing Graphically
Bar Graph: count or percent
Pie Chart: parts of the whole
Stem Plot: shape of distribution
Histogram: great when lots of groups
– Frequency Table
Time Plots
Time Series: measurements of a variable taken
at regular intervals over time
Residual Plots: checking assumptions
Trends, such as seasonal variation
Outliers
‘Extreme’ Values
What do you do with
outliers?
– Ignore them
– Throw them out
– ?
Graphical Examples
Let’s Take a Look
Choosing a Summary
The five-number summary is usually better
than the mean and standard deviation for
describing a skewed distribution or a
distribution with strong outliers.
Use the mean and standard deviation for
reasonably symmetric distributions that are
free of outliers.
Describing Distributions with Numbers
Mean: simple average
– is sensitive to extreme scores
– not necessarily a possible value
To calculate: add the values and divide by the
number of items
Median: middle score
– not sensitive to extreme scores
To Calculate:
– rank data from smallest to largest
– if n is odd, median is the middle score
– if n is even, median is the average of two
middle scores
Mode: most frequent score
– does not always exist
– unstable
– can be used with qualitative data
Measures of Dispersion
(Variability)
Range
– totally sensitive to extreme scores
– easy to compute
To Calculate: high score – low score
Variance: measures squared distances
from the mean
– large values of suggest large variability
Standard Deviation: square root of the variance
Empirical Rule
Should be used for ‘mound shape’ data
– approx. 68% of the data fall between mean +/- SD
– approx. 95% of the data fall between mean +/- 2 *
SD
– approx. 99.7% of the data fall between mean +/- 3
* SD
Let’s give it a try
Let’s use faculty experience.
Why?
What should we do with it?
Quartiles and 5-Number Summary
Quartiles divide ordered numerical data into
four equally sized parts.
– 1st quartile, Q1, 25% below and 75% above
– 2nd quartile, Q2, median, 50% below and 50%
above
– 3rd quartile, Q3, 75% below and 25% above
The low score, Q1, Q2, Q3, and the high
score are known as the five number
summary of a data set.
BoxPlots
Particularly helpful in comparing 2 or more
groups
Box shows central 50% of data and the median
Whiskers show extremes
Let’s give it a try
Let’s use the $ in the pocket data.
Why?
What should we do with it?
1.5 X IQR Criterion
Interquartile Range is
the distance between the
1st and 3rd quartiles
Call an observation a
suspected outlier if it falls
more than 1.5 X IQR
above the 3rd quartile or
below the 1st quartile
Example on page 46
Normal Distributions
Density Curve: can often describe the overall
pattern of a distribution
– Total area of 1 under the curve
– Areas under the curve are relative frequencies
The mean, median, and quartiles can be ‘eyed’
on a density curve.
Normal Distributions
Bell-shaped, symmetric, unimodal curve
The mean and standard deviation completely specify
the normal distribution
N  , 
– Mean is the center of symmetry
– SD is the distance from the mean to the change of
curvature points
Standardizing Observations
The Z-score of an observation gives
the # of standard deviations it is above
or below the mean
x xx
z


s
Standard Normal
Standard Normal is a special
case of the normal where N(0,1)
Let’s do some examples.
We will need to use Table A.