The Normal Distribution

1
The Normal Distribution
2
Frequency Distributions

• Many types of distributions
• Common Distributions
• Normal
• T distribution
• Uniform
• Gamma
• Rayleigh
• F distribution
• Parametric Statistics Assume Normality
• Test for normality

3
Testing for Normality

• Skewness and Kurtosis
• Histogram Plot with normal distribution curve
  superimposed
• Q-Q Plot
• Kolmogorov-Smirnov Test

4
What if my data is not normal?

• Nonparametric procedures
• Log transformation
• Square root transformation
• Transform data into categorical variables

5
Standardization

• Standardizing scores is the process of converting
  each raw score in a distribution to a z score (or
  standard deviation units)
• Raw Score the individual observed scores on
  measured variables

6
Formula for Calculating z Score
OR

• Also known as a standard score
• Helps to understand where a score lies in
  relation to other scores on the distribution
• Indicates how far above or below the mean a given
  score in the distribution is in standard
  deviation units
• Calculated using mean and standard deviation

OR
7
Standard Normal Distribution
8
The Central Limit Theorem

• Central limit theorem As long as you have a
  reasonably large sample size (e.g., n 30), the
  sampling distribution of the mean will be
  normally distributed (i.e., a bell curve) even if
  the distribution of scores in your sample is not
• the sum of a large number of independent
  observations from the same distribution has,
  under certain general conditions, an approximate
  normal distribution. Moreover, the approximation
  steadily improves as the number of observations
  increases.

9
What is a Standard Error?

• A standard error is the standard deviation of the
  sampling distribution of a given statistic (e.g.,
  the mean, the difference between two means, the
  correlation coefficient, etc.)
• It is the measure of how much random variation we
  would expect from equally sized samples drawn
  from the same population
• It is the denominator in the formulas used to
  calculate many inferential statistics

10
Example Standard Errors in Depth

• Imagine that we wanted to find the average shoe
  size of adult American women
• Suppose we selected a random sample of 100 women
  from our population. (Measuring the shoe size of
  all American women is too expensive and tedious.)
• Now suppose that we repeat this process of
  selecting random samples of 100 American women,
  measuring their shoe sizes, and replacing the
  sample in the population.

• This process of randomly sampling, calculating
  the mean and returning the members back to the
  population, is known as sampling with replacement
• All the random samples of the population created
  their own distribution, which is called a
  sampling distribution of the mean

11
Standard Errors in Depth (continued)

• We can plot sample means in frequency graphs to
  form distributions of sample means (just like we
  do with raw scores)
• The mean and standard deviation of the sampling
  distribution of the mean have special names
• The mean of the sampling distribution is called
  the expected value because the mean of the
  sampling distribution of the means is (or
  expected to be) the same as the population mean

• The standard deviation of the sampling
  distribution is called the standard error
• The standard error of the mean provides a measure
  of how much error we can expect when we say that
  a sample mean represents the mean of the larger
  population (hence, it is called the standard
  error)

Expected value

Population mean
Population standard deviation

Standard error
12
How to Calculate the Standard Error

• Since it is costly and difficult to draw several
  samples from a population, you often must make
  due with a single sample. Therefore, it is
  important to examine two characteristics of your
  sample
• The sample size
• The larger the sample, the more likely it will
  represent the population (if chosen randomly)
• The variation of scores within the sample
• If scores in the sample are diverse, we can
  assume that the population is the same, which can
  reduce our confidence that our sample accurately
  represents the population

Population
Sample
Sample
Sample
13
How to Calculate the Standard Error

• The formula is simply the standard deviation of
  the sample (or population) divided by the square
  root of the sample size
• Small samples with large standard deviations
  produce large standard errors
• This makes it difficult to have confidence that
  the sample accurately represents the population
• In contrast, a large sample with a small standard
  deviation will produce a small standard error
• This makes it more likely that the sample
  accurately represents the population

OR
where ? the standard deviation
for the population s the
sample estimate of the
standard deviation n the sample
size
14
The Use of Standard Errors in Inferential
Statistics

• Inferential statistics Statistics generated from
  sample data used to draw conclusions about
  characteristics of a population from which the
  sample was drawn
• Suppose we want to know whether a relationship
  that we find between two variables using sample
  data represents a relationship between the two in
  the larger population
• To answer this question, we need to use standard
  errors

• To summarize, the standard error is used in
  inferential statistics to see whether our sample
  statistic is larger or smaller than the average
  differences (variance or error) in the statistic
  we would expect to occur by chance.

15
Sample Size and Standard Deviation Effects on the
Standard Error
Data collected from a study was examined to
compare the motivational beliefs of 137
elementary school students with 536 middle school
students
Table 6.4 Standard deviations and sample sizes Table 6.4 Standard deviations and sample sizes Table 6.4 Standard deviations and sample sizes Table 6.4 Standard deviations and sample sizes Table 6.4 Standard deviations and sample sizes
Elementary School Sample Elementary School Sample Middle School Sample Middle School Sample
Standard Deviation Sample Size Standard Deviation Sample Size
Expect to do well on test 1.38 137 1.46 536

• Suppose we wanted to know the standard error of
  the mean on the variable I expect to do well on
  the test for each of the two groups in the
  study, the elementary school students and the
  middle school students

• Looking at the Table 6.4, we have the necessary
  statistics to calculate the standard errors for
  each sample (i.e., the standard deviations and
  sample sizes)

16
Sample Size and Standard Deviation Effects on the
Standard Error

• Looking at the Table 6.4, the standard deviations
  look very similar however, there is a large
  difference in the two sample sizes
• As shown earlier in this chapter, to find the
  standard error, we simply divide the standard
  deviation by the square root of the sample size
• For the elementary school sample, we need to
  divide 1.38 by the square root of 137

• Using the same process for the middle school
  sample, we calculate a standard error of 0.06
• Notice that the standard error of the middle
  school sample is half the size of the elementary
  school sample (see next slide)
• This difference was due to the difference in
  sample size, which plays a big role in
  determining the size of the standard error

?

0.12
17
Graph for Sample Size Example
18
Chance, Probability, and Error

• When making inferences from a sample to a
  population (as in inferential statistics), there
  is always some possibility that the sample that
  was selected from the population does not
  accurately represent the population. This is
  where the concepts of chance, probability, and
  error come into play.
• Chance The probability of a statistical event
  occurring due simply to random variations in the
  characteristics of samples of given sizes
  selected randomly from a population
• Error Also known as random sampling error, this
  refers to differences between the sample
  characteristics and the characteristics of the
  larger population caused merely by random
  fluctuations, or variability, involved in the
  process of selecting random samples from a
  population.

When you randomly select two samples of the same
size from the same population, you are likely to
find differences between these two samples.
These differences are due to error, or random
sampling error.
19
Hypothesis Testing

• A hypothesis establishes a criterion that will be
  used to decide whether or not a hypothesis should
  be rejected (e.g., that there is no difference in
  the driving ability of men and women).
• Null Hypothesis (Ho) the hypothesis always
  suggests that there will be no effect in the
  population
• Alternative Hypothesis (HA or H1) An alternative
  to the null hypothesis, it claims that there is
  an effect in the population
• An example of a null hypothesis stating that the
  population mean, µ, will be equal to the mean of
  the sample
• Ho µ
• There are two types of alternative hypotheses
  that can be made. A two-tailed alternative
  hypothesis does not speculate that a sample is
  less than or greater than the population, just
  that it differs. A two-tailed alternative
  hypothesis claiming that the population mean will
  not be equal to the sample mean is denoted as
• HA µ ?
• A one-tailed alternative hypothesis is a
  directional claiming that one value will be
  greater. A one-tailed alternative hypothesis
  claiming that the population mean will be less
  than the sample mean is denoted as
• HA µ lt

20
Errors in Hypothesis Testing
Before deciding whether to reject or retain the
null hypothesis of no effect in the population
the researcher must decide how willing he or she
is to reject the null hypothesis when it is
actually true. In other words, when deciding
that an effect in the sample represents a genuine
phenomenon in the population, one must conclude
that the result was not just due to random
sampling error. We can never be certain that a
result is not due to random sampling error, so
when we reject the null hypothesis we may be
wrong. In the sciences we are usually willing to
live with an error rate of 5, so we set an alpha
level (a) of .05. If the p value is smaller than
the alpha level, the null hypothesis is rejected
(see next slide).

• Type II Error Failing to reject the null
  hypothesis when it is actually false
• To avoid a Type II error a liberal alpha level
  such as .10 can be used

• Type I Error Rejecting the null hypothesis when
  it is actually true
• To avoid a Type I error a conservative alpha
  level like .01 is used.

21
Graphic Demonstrating Hypothesis Testing for a
Two-Tailed Test
22
Statistical Significance

• Statistical significance the probability (p or p
  value) that a statistic derived from a sample
  represents some genuine phenomenon in the
  population. In other words, the effect observed
  in the sample data is not due to random sampling
  error, or chance.
• To determine statistical significance, we must
  compare the size of the effect to our measure of
  random sampling error, which is usually a measure
  of standard error.

23
Effect Size, Statistical Significance, and
Practical Significance

• The Problem Because measures of statistical
  significance rely on the standard error, and the
  standard error is greatly influenced by sample
  size, large sample sizes often produce
  statistically significant results, even for small
  effects.

• Example Comparing a sample mean of 105 with a
  population mean of 100, standard deviation of 15,
  using samples of n 25 and n 1600
• For a 25 person sample
  For a 1600 person sample
• t 1.67
  
  t 13.33
• The p value for a t of 1.67 is between .10 and
  .20 The p value for a t of 13.33 is lt.0001

24
Effect Size, Statistical Significance, and
Practical Significance (continued)

• The cure Effect size
• To deal with this problem of sample size
  affecting statistical significance, statisticians
  calculate effect sizes for their statistics.
  Effect sizes provide a measure of the statistical
  effect while minimizing the role of sample size.
• The effect size is calculated by essentially
  removing the sample size from the standard error.
  This causes the effect to be expressed in
  standard deviation, rather than standard error,
  units.
• Effect sizes provide a measure of practical
  significance, using the following guidelines
• d less than .20 is small
• d between .25 and .75 is moderate
• d greater than .80 is large.
• Because practical significance is subjective it
  is important to take into account the effect size
  and statistical significance, and understand that
  it is easier to get chance results form a small
  sample size than it is from a large sample size

25
Confidence Intervals

• Confidence intervals offer another measure of
  effect size. By using probability and confidence
  intervals a researcher can make educated guesses
  about the approximate value of a population
  parameter.
• Most of the time researchers want to be either
  95 or 99 confident that the confidence interval
  contains the population parameter. Confidence
  intervals are calculated by

26
Confidence Interval Example

• Suppose that we have a random sample of 1000 men.
  We have measured their shoe size and found they
  have a mean shoes size of 10 with a standard
  deviation of 2. The standard error of the mean
  for this sample is .06. Lets calculate a 95
  confidence interval for the population mean.
• First, looking in Appendix B for a two-tailed
  test with df infinity and a .05, we find t95
  1.96.
• Plugging this value into our confidence interval
  formula we, get the following
• CI95 10 (1.96)(.06)
• CI95 10 .12
• CI95 9.88, 10.12

We are 95 confident that the interval between
9.88 and 10.12 contains the population mean.
27
Conclusion
For several decades, statistical significance has
been the measuring stick used by social
scientists to determine whether the results of
their analyses were meaningful. But tests of
statistical significance are quite dependent on
sample size. With large samples, even trivial
effects are often statistically significant,
whereas with small sample sizes, quite large
effects may not reach statistical significance.
Because of this there has been an increasing
appreciation of measures of practical
significance. When determining the practical
significance of results consider all of the
measures at your disposal. Is the result
statistically significant? How large is the
effect size? How wide is the confidence
interval? And in the context of the real world,
how important and meaningful is the statistical
effect?