Descriptive Statistics and Inferential Statistics

1. How to Design and Evaluate Research in Education
(Descriptive Statistics) and (Inferential Statistics)
Chapter 10 & 11
By: Istiqlal Eid

2. Outline of Chapter 10 and 11
• Main Points in Chapter 10 and 11.
• Expansion, themes, and explanation
• Examples and Questions

3. Ice-Breaking Questions
• What are descriptive and inferential statistics (with
examples)?
• Can you use both descriptive and inferential
statistics?
• How do you interpret descriptive statistics?
• How do you interpret inferential statistics?

4. Main Points
• 1. What Is a Descriptive Statistics?
• 1.1. Statistics Versus Parameters.
• 1.2. Nomerical Data.
• 1.3. Quantitive Data.
• 1.4. Categorical Data.
• 2. The Normal Curve.
• 3. The Standard Deviation of a Normal Distribution.
• 4. The Importance of the Normal Curve and z Scores.
• 5. Correlation

5. Descriptive Statistics
•They permit researchers to describe the
information contained in many, many scores
with just a few indices, such as the mean or
median.

6. Statistics Versus Parameters.
• Statistics- Indices ( mean and median) that are
calculated for a sample drawn from a population.
• Parameters- Such indices that are calculated from
the entire population.
• Note: We refer mainly to statistics since most educational research
involves data from samples rather than from populations.

7. Two Types of Numerical Data
Quantitative Data- differ in degree or amount.
They are reported in terms of scores.
(ex- The anxiety scores of all first year students enrolled at university x in
year y).
Categorical Data- differ in kind, but not in degree
or amount. They simply indicate the total number of objects and the
like, a researcher finds in a particular category.

9. Four Types of Measurement Scales

10. •Ordinal, interval and ratio scales pertain to
quantitative data.
•Nominal scale pertains to categorical data.

11. Techniques for Summarizing Quantitative
Data
The techniques are appropriate for use only with quantitative data
not for categorical data.
• Frequency Polygons
- Frequency Distribution
- Grouped Frequency Distribution.
- Frequency Polygon
- Positively/ Negatively Skewed Polygon

12. - Normal Curve
- The Standard Deviation of a Normal Distribution
- Standard Scores and the Normal Curve
- Probability and Z score
- The Importance of the Normal Curve and z Scores
- -Correlation/ Correlation Coefficients
- Scatterplots

13. Techniques for Summarizing Categorical
Data
•The Frequency Table
•Par Graphs and Par Charts
•The Crossbreak Table

14. The Normal Curve
• Many distribution of data tend to follow a certain shape of
distribution curve- a normal distribution.
• The large majority of the scores are concentrated in the middle, and
the scores decrease the farther away from the middle they are.

15. The Standard Deviation of a Normal Curve
• SD is the most useful index of variability, and it is a single number that
represents the spread of a distribution.
• SD can be calculated as this:
Where SD is the symbol for standard deviation, Σ is the symbol for “sum of,”
X is the symbol for a raw score, X – is the symbol for the mean, and n
represents the number of scores in the distribution.

16. The Importance of the Normal Curve and
z Scores
• The use of z scores in relation to the normal curve is always qualified when
the distribution of scores is normal.
• Z score can be calculated regardless of the shape of the distribution of
original scores. But it is only when the distribution is normal that the
conversion to percentages or probabilities is legitimate.

17. Correlation
• In correlational research, researchers seek to determine whether a
relationship exists between two variables. Sometimes, such
relationships are useful in prediction.
• Most often the eventual goals is to say something about causation.

18. How to Design and Evaluate Research in Education
(Inferential Statistics)
Chapter 11

19. Ice-Breaking Questions
• “Survey results show that 60 percent of those we interviewed
support our bon issue”.
So, the question is:
Can we be sure then that 60 percent of the voters will go for it?

20. Answer of ice breaking question
No, not exactly. But it is very likely that
between 55 percent and 65 percent will!

21. Ice-Breaking Questions
• “Hypotheses can never be proven, only supported”. Is this
statement true or not? Explain.
• No two samples will be the same in all of their
characteristics. Why won’t they?

22. Main Points
• What are Inferential Statistics ?
• Sampling Error
• The Distribution of Sample Means
• Standard Error of the Mean
• Estimating the Standard Error of the Mean
• Confidence Intervals

23. Main Points
• Hypothesis Testing
• Significance Levels
• Confidence Intervals and Probability
• Comparing More than One Sample
• The standard Error of the Difference Between Sample Means
• The Null Hypothesis

24. Main Points
• Hypothesis Testing: A Review
• Practical Versus Statistical Significance
• Inference Techniques
• Tests of Statistical Significance
• Parametric Tests for Analyzing Quantitative Data
• Parametric Tests for Analyzing Categorical Data

25. Main Points
• Nonparametric Tests for Analyzing Quantitative Data
• Nonparametric Tests for Analyzing Categorical Data
• Summary of Techniques
• Power of a Statistical Test
• Can Statistical Power Analysis be Misleading?

26. What are Inferential Statistics ?
• Inferential statistics refer to certain procedures that allow
researchers to make inferences about a population based on data
obtained from a sample.
• The term "probability," as used in research, refers to the
predicted relative frequency with which a given event will occur.

27. Sampling Error
• The term "sampling error" refers to the variations in
sample statistics that occur as a result of repeated
sampling from the same population.

28. The Distribution of Sample Means
• A sampling distribution of means is a frequency distribution resulting
from plotting the means of a very large number of samples from the
same population.
• The standard error of the mean is the standard deviation of a
sampling distribution of means. The standard error of the difference
between means is the standard deviation of a sampling distribution
of differences between sample means.

29. Standard Error of the Mean
(SEM)
• It is the standard deviation of a sampling distribution of means.
• If we know or can accurately estimate the mean and the standard deviation
of the sampling distribution, we can determine whether it is likely or unlikely
that a particular sample mean could be obtained from that population.

30. Estimating the Standard Error of the Mean
• We cannot calculate the standard error of the mean directly, since we would
need, literally, to obtain a huge number of samples and their means.
• However, the standard error can be calculated using a simple formula
requiring the standard deviation (SD)of the population and the size of the
sample. SD of the population can be estimated using the SD of the sample.
• So, SEM =

31. Confidence Intervals
• A confidence interval is a region extending both above and below a sample
statistic (such as a sample mean) within which a population parameter (such
as the population mean) may be said to fall with a specified probability of
being wrong.

32. Hypothesis Testing
• Statistical hypothesis testing is a way of determining the probability
that an obtained sample statistic will occur, given a hypothetical
population parameter.
• A research hypothesis specifies the nature of the relationship the
researcher thinks exists in the population.
• The researcher formulates both a research hypothesis and a null
hypothesis.

33. Significance Levels
• The term "significance level" (or "level of significance"), as used in
research, refers to the probability of a sample statistic occurring as a
result of sampling error.
• The significance levels most commonly used in educational research
are the .05 and .01 levels.
• Statistical significance and practical significance are not necessarily
the same. Just because a result is statistically significant does not
mean that it is practically (i.e., educationally) significant.

34. Comparing More than One Sample
• A researcher might want to determine if there is a difference in attitude
between two groups.
• We ask about a difference between means: Is the difference we have found
a likely or unlikely occurrence?
• It is possible that the difference can be attributed simply to sampling error-
to the fact that certain samples, rather than others, were selected ( the “luck
of the draw”, so to speak). Inferential statistics help out.

35. The Standard Error of the Difference
Between Sample Means
• Differences between sample means are also likely to be normally distributed.
• The distribution has its own mean and standard deviation.
• The mean of the sampling distribution of differences between sample means
is equal to the difference between the means of the two populations.
The SD of this distribution is called standard error of the difference (SED).
SED=
1 and 2 refer to the respective samples

36. The Null Hypothesis
• To test the research hypothesis, the researcher must formulate a null
hypothesis.
• The null hypothesis typically specifies that there is no relationship in
the population (the difference between the means of the two
populations is zero).

37. Hypothesis Testing: A Review
1. State the research hypothesis.
2. State the null hypothesis.
3. Determine the sample statistics pertinent to the hypothesis
4. Determine the probability of obtaining the sample results if the null
hypothesis is true.
5. If the probability is small, reject the null hypothesis, thus affirming the research
hypothesis.
6. If the probability is large, do not reject the null hypothesis, which means you cannot
affirm the research hypothesis.

38. Practical Versus Statistical Significance
• Statistical significance only means that one’s results are likely to occur
by chance less than a certain percentage of the time, say 5 percent.
• Whenever we have a large enough random sample, almost any result
will turn out to be statistically significant.
• A very small correlation coefficient may turn out to be statistically
significant but have little (if any) practical significance.

39. Inference Techniques
• The type of data a researcher collects often influences the type of statistical
analysis required.
• A statistical technique appropriate for quantitative data will generally be
inappropriate for categorical data.
• Two basic types of inference techniques that researchers use:
- Parametric techniques.
- Nonparametric techniques.

40. Tests of Statistical Significance
• A one-tailed test of significance involves the use of probabilities based on
one-half of a sampling distribution because the research hypothesis is a
directional hypothesis.
• A two-tailed test, on the other hand, involves the use of probabilities based
on both sides of a sampling distribution because the research hypothesis is a
nondirectional hypothesis.

41. Parametric Tests for Analyzing Quantitative
Data
• A parametric statistical test requires various kinds of assumptions about the
nature of the population from which the samples involved in the research
study were taken.
• Some of the commonly used parametric techniques for analyzing
quantitative data include the t-test for means, ANOVA, ANCOVA,
MANOVA, and the t-test for r.

42. Parametric Tests for Analyzing Categorical
Data
• The most common parametric technique for
analyzing categorical data is the t-test for differences
in proportions.

43. Nonparametric Tests for Analyzing
Quantitative Data
• A nonparametric statistical technique makes few, if any, assumptions about
the nature of the population from which the samples in the study were
taken.
• Some of the commonly used nonparametric techniques for analyzing
quantitative data are the Mann-Whitney U test, the Kruskal-Wallis one-way
analysis of variance, the sign test, and the Friedman two-way analysis
variance.

44. Nonparametric Tests for Analyzing
Categorical Data
• The chi-square test is the nonparametric technique most commonly used to
analyze categorical data.
• The contingency coefficient is a descriptive statistic indicating the degree of
relationship that exists between two categorical variables.

45. Summary of Techniques
• The details of both mathematical rationale and calculation differ greatly
among these technique, the most important things to remember are:
1- The end product of all inference procedures is the same: a statement of
probability relating the sample data to hypothesized population characteristics.
2- All inference techniques assume random sampling. Without random
sampling, the resulting probabilities are in error- to an unknown degree.
3- Inference techniques are intended to answer only one question: Given the
sample data, what are probable population characteristics?

46. Summary of Techniques
• The above techniques do not help decide
whether the data show results that are
meaningful or useful.
• They indicate only the extent to which they
may be generalizable.

47. Power of a Statistical Test
• The power of a statistical test for a particular set of data is the likelihood of
identifying a difference between population parameters when it in fact exists.
• Parametric tests are generally, but not always, more powerful than
nonparametric tests.

48. Can Statistical Power Analysis be
Misleading?
• There was a study on federal ban on assault weapons ,which found
that gun homicides declined 6.7 percent more than expected!
• This difference was not statistically significant.
• The authors concluded that a larger sample and longer time period
were needed to have a sufficiently powerful statistical test.
• The question is: Are such studies worth doing?