Adopted from ;Merryellen Towey Schulz, Ph.D.
College of Saint Mary
EDU 496

• Last year’s enrollment
• Collections of figures numerical data
• Average enrollment
• Summary measures per month last year calculated from a collection of data
• Activity of using and interpreting a collection of
• Evaluators made a projection of next year’s enrollments numerical data

• Use of numerical information to summarize, simplify, and present data.
• Organized and summarized for clear presentation
• For ease of communications
• Data may come from studies of populations or samples

Design
Survey Studies
Meta-analysis
Causal comparative studies
Experimental
Descriptive Statistics
Percentages, measures of central tendency and variation
Effect sizes
Measures of central tendency & variation, percentages, standard scores
Measures of central tendency & variation, percentages, standard scores, effect sizes

• Central tendency
• Mode
• Median
• Mean
• Variation
• Range
• Standard deviation
• Normal distribution

• Standard score
• Effect size
• Correlation
• Regression

• To generalize or predict how a large group will behave based upon information taken from a part of the group is called and
INFERENCE
• Techniques which tell us how much confidence we can have when we
GENERALIZE from a sample to a population

• Hypothesis
• Null hypothesis
• Alternative hypothesis
• ANOVA
• Level of significance
• Type I error
• Type II error

Descriptive Statistics Inferential Statistics
• Graphical
– Arrange data in tables
– Bar graphs and pie charts
• Numerical
– Percentages
– Averages
– Range
• Relationships
– Correlation coefficient
– Regression analysis
• Confidence interval
• Margin of error
• Compare means of two samples
– Pre/post scores
– t Test
• Compare means from three samples
– Pre/post and follow-up
– ANOVA = analysis of variance

• Sampling Error
– Inherent variation between sample and population
– Source is “chance or luck”
– Results in bias
• Sample statistic -- a number or figure
– Single measure -- how sure accurate
– Comparing measures --see differences
• How much due to chance?
• How much due to intervention?

(Significant) ?
• Statistics, descriptive or inferential are NOT a substitute for good judgment
– Decide what level or value of a statistic is meaningful
– State judgment before gathering and analyzing data
• Examples:
– Score on performance test of 80% is passing
– Pre/post rules instruction reduces incidents by 50%

• Population Measure (statistic)
– There is no sampling error
– The number you have is “real”
– Judge against pre-set standard
• Inferential Measure (statistic)
– Tells you how sure (confident) you can be the number you have is real
– Judge against pre-set standard and state how certain the measure is

Statistics has two major chapters:
• Descriptive Statistics
• Inferential statistics

Descriptive Statistics
• Gives numerical and graphic procedures to summarize a collection of data in a clear and understandable way
Inferential Statistics
• Provides procedures to draw inferences about a population from a sample

• Central Tendency measures . They are computed to give a “center” around which the measurements in the data are distributed.
• Variation or Variability measures . They describe “data spread” or how far away the measurements are from the center.
• Relative Standing measures . They describe the relative position of specific measurements in the data.

Measures of Central Tendency
• Mean:
Sum of all measurements divided by the number of measurements.
• Median:
A number such that at most half of the measurements are below it and at most half of the measurements are above it.
• Mode:
The most frequent measurement in the data.

Measurements Deviation x x - mean
3 -1
0
4
6
7
1
7
5
5
2
-4
0
2
3
-3
3
1
1
-2
40 0
• MEAN = 40/10 = 4
• Notice that the sum of the
“deviations” is 0.
• Notice that every single observation intervenes in the computation of the mean.

Measurements Measurements x
3
Ranked x
0
2
6
7
0
4
1
7
5
5
1
2
3
4
5
5
6
7
7
40 40
• Median: (4+5)/2 =
4.5
• Notice that only the two central values are used in the computation.
• The median is not sensible to extreme values

Measurements
2
6
7
0
4
1
7
5
5 x
3 • In this case the data have tow modes:
• 5 and 7
• Both measurements are repeated twice

Measurements x
3
5
1
1
3
8
4
7
3
• Mode: 3
• Notice that it is possible for a data not to have any mode.

• Steps:
– Compute each deviation
– Square each deviation
– Sum all the squares
– Divide by the data size (sample size) minus one: n-1

Measurements Deviations Square of deviations
5
5 x
3
1 x - mean
-1
1
1
-3
1
1
1
9
7
0
4
7
2
6
3
-2
2
3
-4
0
9
4
4
9
16
0
40 0 54
• Variance = 54/9 = 6
• It is a measure of
“spread”.
• Notice that the larger the deviations (positive or negative) the larger the variance

• It is defines as the square root of the variance
• In the previous example
• Variance = 6
• Standard deviation = Square root of the variance = Square root of 6 = 2.45

• The p-the percentile is a number such that at most p% of the measurements are below it and at most 100 – p percent of the data are above it.
• Example, if in a certain data the 85 th percentile is 340 means that 15% of the measurements in the data are above 340. It also means that 85% of the measurements are below 340
• Notice that the median is the 50 th percentile

• At least 75% of the measurements differ from the mean less than twice the standard deviation.
• At least 89% of the measurements differ from the mean less than three times the standard deviation.
Note:
This is a general property and it is called Tchebichev’s Rule: At least 1-1/k 2 of the observation falls within k standard deviations from the mean. It is true for every dataset.

Suppose that for a certain data is :
• Mean = 20
• Standard deviation =3
Then:
• A least 75% of the measurements are between 14 and 26
• At least 89% of the measurements are between 11 and 29

• When the Mean is greater than the Median the data distribution is skewed to the Right.
• When the Median is greater than the Mean the data distribution is skewed to the Left.
• When Mean and Median are very close to each other the data distribution is approximately symmetric
.