Descriptive & Inferential Statistics

Adopted from ;Merryellen Towey Schulz, Ph.D.

College of Saint Mary

EDU 496

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

• Evaluators made a projection of next year’s enrollments

Descriptive Statistics• 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

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

Experimental Measures of central tendency & variation, percentages, standard scores, effect sizes

Descriptive Stats Vocabulary• Central tendency

• Mode

• Median

• Mean

• Variation

• Range

• Standard deviation

• Normal distribution

Descriptive Stats Vocabulary cont’d

• Standard score

• Effect size

• Correlation

• Regression

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 GENERALIZE from a sample to a population

Inferential Stats Vocabulary

• Hypothesis

• Null hypothesis

• Alternative hypothesis

• ANOVA

• Level of significance

• Type I error

• Type II 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 variance

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?

• How much due to intervention?

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

– Pre/post rules instruction reduces incidents by 50%

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– Judge against pre-set standard and state how

certain the measure is

Descriptive Statisticsfor one variable

Statistics has two major chapters:

• Descriptive Statistics

• Inferential statistics

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

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 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.

Example of Mean

Measurements Deviationx x - mean3 -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.

• Notice that every single observation intervenes in the computation of the mean.

Example of Median

• 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 Measurements Ranked

x x3 05 15 21 37 42 56 57 60 74 7

40 40

Example of Mode

Measurements

x3551726704

• In this case the data have tow modes:

• 5 and 7• Both measurements are

repeated twice

Example of Mode

Measurementsx351147383

• Mode: 3

• Notice that it is possible for a data not to have any mode.

Variance (for a sample)

• Steps:– Compute each deviation– Square each deviation– Sum all the squares– Divide by the data size (sample size)

minus one: n-1

Example of Variance

Measurements Deviations Square of deviations

x x - mean3 -1 15 1 15 1 11 -3 97 3 92 -2 46 2 47 3 90 -4 164 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 variance

The standard deviation

• 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

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

• Notice that the median is the 50th percentile

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 from the mean. It is true for every dataset.

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

• At least 89% of the measurements are between 11 and 29

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 approximately symmetric.