Descriptive Statistics

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Descriptive Statistics

• Descriptive Statistics describe a set of data.

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• If I want to report findings from an observation
  or a survey, I may want to use a frequency
  distribution.
• A frequency distribution may be a simple chart, a
  list, or a graph.
• A graph of information always plots the frequency
  along the y-axis, and the subject of the graph
  along the x-axis.

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Measures of Central Tendency

• Measures of central tendency provide statistics
  that indicate the average or typical score in the
  distribution.
• Mean
• Median
• Mode

• There are three measures of central tendency

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Mean

• The mean is the arithmetic average of all the
  scores in the distribution. It is calculated by
  adding all the scores in the distribution and
  then dividing this sum by the number of scores.

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Median
The median is in the middle of the road!

• The median is the middle score of the
  distribution, the point that divides a
  rank-ordered distribution into halves containing
  an equal number of scores. Thus 50 of the scores
  lie below the median and 50 lie above the
  median.

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Mode

• The mode is simply the score in the distribution
  that occurs most frequently.

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Graphing Measures of Central Tendency

• When graphing the mean, median and mode of a
  distribution, roughly speaking, a distribution
  has positive skew if the right tail is longer and
  negative skew if the left tail is longer.

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Positively Skewed

• This distribution has a positive skew. Note that
  the mean is larger than the median.

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• IE. In a neighborhood of relatively low incomes,
  a few millionaires move in. Those few high
  salaries will inflate the mean (average), but the
  median will remain relatively low.

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Negatively Skewed

• This distribution has a negative skew. The median
  is larger than the mean.

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• IE. In a particular well-to-do neighborhood, a
  few low-income residents move in. The overall
  average income will drop a bit, but the median
  will remain relatively high.

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Measures of Variability

• Measures of variability show how spread out the
  distribution of scores is from the mean, or how
  much dispersion or scatter exists in the
  distribution. If there is a large degree of
  dispersion, that is, if the scores are very
  dissimilar, we say the distribution has a large
  or high variability, or variance. If the scores
  are very similar, there is a small degree of
  dispersion and a small variance.

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Measures of Variability

• Range
• Standard Deviation

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Range

• The range is simply the numerical difference
  between the highest and lowest scores in the
  distribution.

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Standard Deviation

• The measure of variability used most often in
  research is the standard deviation, a statistic
  that indicates the average distance of the scores
  from the mean of the distribution.

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• IE. Our class took Unit Exam 2.
• I scored a 76.
• How did I do compared to everyone else in class?
• I need to figure out what the class average was,
  figure out the standard deviation from the mean,
  and Ill know how well I did.

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Standard Deviation
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With a mean of 62 and a SD of 3, 95 of scores
should fall between what scores?
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15.Which of the following sets of numbers has the
largest standard deviation? (a) 2, 1, 0, 1,
2 (b) 1.00, 1.25, 1.50, 1.75, 2.00, 2.25, 2.50,
2.75, 3.00 (c) 2, 6, 10, 14, 18 (d) 5.756, 5.765,
5.890, 5.895, 5.923 (e) 91, 92, 93, 94, 95
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Graphing Standard Deviation

• Find the mean of your distribution set.
• Calculate the SD on your calculator.
• The mean is set at 0.
• 1 and -1 are your SD above and below the mean.
• IE. Your mean is 56 with a SD of 6. 1 would be
  62, and -1 would be 50.
• Calculate - 2 and - 3 in the same manner.

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Graphing Standard Deviation

• What does this tell us?
• If the mean of a set of class scores on a unit
  exam was 72, with a SD of 8, 68 of students
  scored between a 64 and an 80. Your score of a
  76 would be close to being better than 68 of the
  rest of the class.
• Approximately 95 of the class scored between a
  56 and an 88. Your score of a 50 would indicated
  that roughly 96 of the class did better than you
  on the test.

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Graphing Data

• Scatterplot A graphed cluster of dots that
  represent the values of two variables.

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• The SLOPE of the points suggests whether there is
  a positive, negative, or non-existent
  relationship between two variables.

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• POSITIVE CORRELATION as one set of scores
  increases, so does the other

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• NEGATIVE CORRELATION as one set of scores goes
  up, another set goes down

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• NO CORRELATION

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• No correlation relates to a score of 0.00
• A positive correlation ranges from 0.00 to 1.00
• A negative correlation ranges from 0.00 to -1.00

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• How closely the dots are to each other along the
  line indicates the strength or weakness of the
  correlation as well