Descriptive & Inferential
Statistics
Adopted from ;Merryellen Towey Schulz, Ph.D. College of Saint Mary
The Meaning of Statistics
Several Meanings
• Collections of
numerical data
• Summary measures
calculated from a
collection of data
• Activity of using and
interpreting a
collection of numerical
data
• Last year’s enrollment
figures
• Average enrollment
per month last year
Descriptive
Statistics
• Use of numerical information to
summarize, simplify, and present data.
• Organized and summarized for clear
presentation
• For ease of communications
Descriptive Statistics Associated
with Methods and Designs
Design Descriptive Statistics
Survey Studies Percentages, measures of central tendency and variation
Meta-analysis Effect sizes
Causal comparative studies Measures of central tendency & variation, percentages, standard scores
Descriptive Stats Vocabulary
• Central tendency
• Mode
• Median
• Mean
• Variation
• Range
Descriptive Stats Vocabulary
cont’d
• Standard score
• Effect size
Inferential Statistics
• 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
Inferential Stats Vocabulary
• Hypothesis
• Null hypothesis
• Alternative hypothesis
• ANOVA
• Level of significance
• Type I error
Examples of Descriptive and
Inferential Statistics
Descriptive Statistics
• Graphical
– Arrange data in tables – Bar graphs and pie charts
• Numerical
– Percentages – Averages – Range
• Relationships
– Correlation coefficient – Regression analysis
Inferential Statistics
• 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
Problems With Samples
• 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?
What Is Meant By A Meaningful
Statistic
(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
Interpretation of Meaning
• 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
Statistics has two major
chapters:
• Descriptive Statistics
Statistics
Descriptive Statistics
• Gives numerical and graphic procedures to summarize a
collection of data in a clear and
understandable way
Inferential Statistics • Provides
Descriptive Measures
•
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 specificMeasures 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:
Example of Mean
Measurements Deviation x x - mean
3 -1 5 1 5 1 1 -3 7 3 2 -2 6 2 7 3 0 -4 4 0 40 0
•
MEAN = 40/10 = 4
• Notice that the sum of the “deviations” is 0.
Example of Median
• Median: (4+5)/2 = 4.5
• Notice that only the two central values are used in the
computation.
Example of Mode
Measurements x 3 5 5 1 7 2 6 7 0 4• In this case the data have tow modes:
• 5 and 7
Example of Mode
Measurementsx
3 5 1 1 4 7 3 8 3
• Mode: 3
Variance (for a sample)
•
Steps:
– Compute each deviation
– Square each deviation
– Sum all the squares
Example of Variance
Measurements Deviations Square of deviations x x - mean
3 -1 1
5 1 1
5 1 1
1 -3 9
7 3 9
2 -2 4
6 2 4
7 3 9
0 -4 16
4 0 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
The standard deviation
• It is defines as the square root of the
variance
• In the previous example
• Variance = 6
Percentiles
• 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 85th 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
For any data
• 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/k2 of the observation falls within k standard deviations
Example of Tchebichev’s Rule
Suppose that for a certain data is : • Mean = 20
• Standard deviation =3
Then:
• A least 75% of the measurements are between 14 and 26
Further Notes
• 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
