Measures of Central Tendency

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Measures of Central Tendency

• A way to summarize a variable is to identify the
  most typical score.
• Measures of Central Tendency are different
  typical scores and represent values around
  which others concentrate.

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The Mode

• The mode is the value with the highest frequency
  count.
• While the mode can be useful, it is very
  informative as it only tells us which value
  occurred the most frequently.

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The Median

• The median that marks the midpoint. ½ of the
  values exceed the median and ½ of the values are
  below the median.
• The median is better than the mode in summarizing
  a variable, but tells us little about the
  distribution of values.

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The Mean

• The mean incorporates all of the values for a
  variable! Called the average, it is obtained
  by adding up all the values dividing that sum
  by the total number of cases.
• Usually the most accurate, stable and useful
  measure of central tendency.it is vulnerable to
  distortions by extremely high or low values.

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Measures of variability

• A quick measure of variability is the range. To
  calculate the range (R) you subtract the lowest
  value (L) for a variable from the highest value
  (H).
• Although better than nothing, it is based only on
  the two most extreme variable scores.and ignores
  all the other information about the dispersion in
  the data.
• The variance reflects the sum of deviations of
  each value from the mean and provide us with an
  average amount of dispersion or variability.
  The standard deviation is the square root of the
  variance.

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From descriptive to inferential statistics.

• The normal distribution and z-scores connect
  descriptive statistics to inferential statistics.
• The ND adds to the interpretation of the mean and
  standard deviation, and is the basis for
  statistical estimation, hypothesis testing, and
  measures of association.

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Why is the normal curve so important in
statistics?

• A large number of real world variables are
  normally distributed.
• The sampling distribution of several statistics
  are normally distributed.
• The above theorems are why researchers can make
  accurate inferences about populations from the
  analysis of samples. They can accurately
  estimate population parameters using sample
  statistics.

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Key aspects of the normal curve

• The normal curve is symmetrical or bell-shaped.
• The mean is also the most frequently occurring
  value (the mode), and the value that splits the
  distribution in half (the median).
• Assuming a variable is normally distributed in
  the population, we can say much more about the
  standard deviation.

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From Statistical Estimation to Confidence
Intervals.

• Probability theory states that in repeated
  sampling, the distribution of sample means will
  be normally distributed with a mean equal to the
  true population mean.
• Thus, we can define a range of values within
  which the true population mean is likely to be.
• And we can estimate the probability that our
  range includes the population meanwhen we do
  this we have calculated a confidence interval.

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Hypothesis Testing t-tests

• If we are interested in seeing if the means or
  proportions of a variable differ between two
  groups, the t-ratio or t-test is our statistic of
  choice.

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Hypothesis Testing ANOVA

• If you are asked to look for significant
  differences in the means of three or more groups
  your method of choice is analysis of variance
  (ANOVA).
• SSt Total Sum of Squares Total Variation in
  the data.
• SSb Between Group Sum of Squares Between
  group variation in the data.
• SSw Within Group Sum of Squares Within group
  variation in the data.
• The F-ratio represents the size of the
  differences between the groups we are comparing
  relative to the size of the differences within
  the groups we are comparing.

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Hypothesis Testing Chi-square

• When you are testing for differences with
  variables at the nominal or ordinal level, the
  Chi-square statistic is appropriate.
• Chi-square tests if the difference between
  observed frequencies (fo) and expected
  frequencies (fe) is so large it cannot be due to
  chance or random sampling error.

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CORRELATION REGRESSION

• The higher the correlation between X and Y, the
  better our predictions of Y using X will be.
• The value r² measures how much better our
  predictions of Y using X are.if we square
  Pearsons r .85, we obtain r² .72. This is
  the coefficient of determination and means that
  if we use X to predict Y we will improve the
  accuracy of our predictions by 72.
• The coefficient of non-determination is 1- r²,
  and it is the percentage of variability in Y not
  explained by X (due to other causal variables).