Measures of Central Tendency

1
Measures of Central Tendency
to be or not to be Normal
2
TOPICS

• Normal Distributions
• Skewness Kurtosis
• Normal Curves and Probability
• Z- scores
• Confidence Intervals
• Hypothesis Testing
• The t-distribution

3
Is this normal ?
4
Normal Distributions

• Are your curves normal?
• Why do we care about normal curves?
• What do normal curves tell us?
• Answer
• The curves tell us something about the
  distribution of the population
• The curves allow us to make statistical
  inferences regarding the probability of some
  outcomes within some margin of error

5
The normal distribution

• A distribution is easily depicted in a graph
  where the height of the line determined by the
  frequency of cases for the values beneath it.
• Most cases cluster near the middle of a
  distribution if close to normal

6
The Normal Curve

• Bell-shaped distribution or curve
• Perfectly symmetrical about the mean.
• Mean median mode
• Tails are asymptotic closer and closer to
  horizontal axis but never reach it.

7
Skewness and Sample Distributions
Not all curves are normal, even if still
bell-shaped
8
Skewness

• Formula for skewness

9
Kurtosis (Its not a disease)

• Beyond skewness, kurtosis tells us when our
  distribution may have high or low variance, even
  if normal.
• The kurtosis value for a normal distribution will
  equal 3. Anything above this is a peaked value
  (low variance) and anything below is platykurtic
  (high variance).

10
Back to normal distributions

• The power of normal distributions, or those close
  to it, is that we can predict where cases will
  fall within a distribution probabilistically.
• For example, what are the odds, given the
  population parameter of human height, that
  someone will grow to more than eight feet?
• Answer, likely less than a .025 probability

11
Sample Distribution

• What does Andre the Giant do to the sample
  distribution?
• What is the probability of finding someone like
  Andre in the population?
• Are you ready for more inferential statistics?
• Answer Oh boy, yes!!

12
Normal Curves and probability

• We have answered the question of what Andre and
  the Sumo wrestler would do to the distribution
• But what about the probability of finding someone
  the same height as Andre in the population?
• What is the probability of finding someone the
  same height as Dr. Peña or Dr. Boehmer?

13
More on normal curves and probability
Andre would be here
Dr. Boehmer would be here
14
Z-Scores (no sleeping!!)

• We can standardize the central tendency away from
  the mean across different samples with z-scores.
• The basic unit of the z-score is the standard
  deviation.

15
We can use the z-score to score each observation
as a distance from the mean. How far is a given
observation from the mean when its z-score
2? Answer 2 standard deviations. Approximately
what percentage of cases is a given case higher
than if its z-score 2? Answer 97
16
Random Sampling Error

• Ever hear a poll report a margin of error? What
  is that?
• Random Sampling Error standard deviation/
  square root of the sample size
• Or

As the variance of the population increases, so
does the chance that a sample could not reflect
the population parameters
17
Standard Error

• We often refer to both the random sampling error
  with both the chance to err when sampling but
  also the error of a specific sample statistic,
  the mean. We typically use the term Standard
  Error.
• A sample statistic standard error is the
  difference between the mean of a sample and the
  mean of the population from which it is drawn.

18
Standard Error

• Example What if most humans were 200 pounds and
  only 1 million globally were 250 pounds?
• The random sampling error would be low since the
  chance of collecting a sample consisting heavily
  of those heavier humans would be unlikely. There
  would not be much error in general from sampling
  because of the low variance.

19
Standard Error

• Example continued. Now, when we take a sample,
  each sample has a mean. If a population has low
  variance, so should the samples. We should see
  this reflected in low standard error in the mean
  of the sample, the sample statistic.
• Of course, higher variance in the population also
  causes higher error in samples taken from it.

20
Some more notation
Distributions Mean Standard Dev.
Sample of observed data s
Population µ s
Repeated Sampling µ
Random Sampling Error
Error in a Samples mean is the Standard Error
21
Central Limit Theorem

• Remember that if we took an infinite number of
  samples from a population, the means of these
  samples would be normally distributed.
• Hence, the larger the sample relative to the
  population, the more likely the sample mean will
  capture the population mean.

22
Confidence Intervals

• We can actually use the information we have about
  a standard deviation from the mean and calculate
  the range of values for which a sample would have
  if they were to fall close to the mean of the
  population.
• This range is based on the probability that the
  sample mean falls close to the population mean
  with a probability of .95, or 5 error.

23
How Confident Are You?

• Are you 100 sure?
• Social scientists use a 95 as a threshold to
  test whether or not the results are product of
  chance.
• That is, we take 1 out of 20 chances to be wrong
• What do you MEAN?
• We build a 95 confidence interval to make sure
  that the mean will be within that range

24
Confidence Interval (CI)
Y mean Z Z score related with a 95 CI s
standard error
25
Building a CI

• Assume the following

26
CI
Why do we use 1.96?
27
Calculating a 95 CI

1. Lets look at the class population distribution
   of height
2. Is it a normal or skew distribution?
3. Lets build a 95 CI around the mean height of
   the class

28
Why do we care about CI?

• We use CI interval for hypothesis testing
• For instance, we want to know if there is an
  income difference between El Paso and Boston
• We want to know whether or not taking class at
  Kaplan makes a difference in our GRE scores

29
Mean Difference testing
Mean USA
Boston
Las Cruces
El Paso
Income levels
30
(No Transcript)