PSY 360

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PSY 360

• Describing Middle
• Describing Spread

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Description With Statistics

• Aspects or characteristics of data that we can
  describe are
• Middle
• Spread
• Skewness
• Kurtosis
• Statistics that measure/describe middle are mean,
  median, mode
• Statistics that measure/describe spread are
  range, variance, standard deviation, midrange

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Description With Statistics

• Middle central tendency, location, center
• Measures of middle are mean, median, mode
  (keywords)
• Spread variability, dispersion
• Measures of spread are range, variance, standard
  deviation, midrange (keywords)
• Skewness departure from symmetry
• Positive skewness tail (extreme scores) in
  positive direction
• Negative skewness tail (extreme scores) in
  negative direction
• Kurtosis peakedness relative to normal curve

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(No Transcript)
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Skewness
Positive Skewness
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Skewness
Positive Skewness
Negative Skewness
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Kurtosis
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Description With Statistics
S

• Another name for middle is central tendency,
  location, or center
• Mean describes/measures middle
• Another name for spread is variability or
  dispersion
• Variance describes/measures spread
• Why is mean not a correct alternative name for
  middle? Because mean is a statistic and the name
  mean is a reserved keyword
• Why is variance not a correct alternative name
  for spread? Because variance is a statistic and
  the name variance is a reserved keyword

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Describing the Middle of Data

• Another name for middle is
• _________.
• Middle is the aspect of data
• we want to describe.
• We describe/measure the middle of data in a
  sample with the statistics
• Mean.
• Median.
• Mode.
• We describe/measure the middle of data in a
  population with the parameter ? (mu) we
  usually dont know ?, so we estimate it with X.

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Sample Mean

• The sample mean is the sum of the scores divided
  by the number of scores, and is symbolized by
  X-bar, X ?X
• N
• For example 4, 1, 7, N3, ?X12 and X ?X/N
  12/3 4
• Characteristics
• X is the balance point

• ?(X-X)0
• X Minimizes ?(X-X)2 (Least Squares criterion)
• X is pulled in the direction of extreme scores

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Sample Mean

• What is the mean for the following data 4, 1, 7,
  6
• N4
• ?X18
• X ?X/N 18/4 4.5

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Sample Median

• The median is the middle of the ordered
  scores, and is symbolized as X50.
• Median position (as distinct from the median
  itself) is (N1)/2 and is used to find the
  median.
• Find the median of these scores 4, 1, 7
• N3.
• Median position is (31)/2 4/2 2.
• Place the scores in order 1, 4, 7.
• X50 is the score in position/rank 2.
• So X50 4.

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Sample Median

• Another example 4, 1, 7, 6
• N4.
• Median position is (N1)/2 (41)/2 5/2 2.5.
• Place the scores in order 1, 4, 6, 7.
• X50 is the score in position/rank 2.5.
• So X50 (46)/2 10/2 5.
• Characteristics
• Depends on only one or two middle values.
• For quantitative data when distribution is
  skewed.
• Minimizes ?X-X50.

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Sample Mode

• The mode is the most frequent score.
• Examples
• 1 1 4 7, the mode is 1.
• 1 1 4 7 7, there are two modes, 1 and 7.
• 1 4 7, there is no mode.
• Characteristics
• Has problems more than one, or none maybe not
  in the middle little info regarding the data.
• Best for qualitative data, e.g. gender.
• If it exists, it is always one of the scores.
• Is rarely used.

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Describing the Spread of Data

• Another name for spread is _________.
• Spread is the aspect of data we want to
  describe.
• Any statistic that describes/measures spread
  should have these characteristics it should
• Equal zero when the spread is zero.
• Increase as spread increases.
• Measure just spread, not middle.

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Describing the Spread of Data

• We describe/measure the spread of data in a
  sample with the statistics
• Range high score-low score.
• Midrange, MR.
• Sample variance, s².
• Sample standard deviation, s.
• Unbiased variance estimate, s².
• Standard deviation, s.
• We describe/measure the spread of data in a
  population with the parameter ? (sigma) or ?²
  we usually dont know ? or ?², so we estimate
  them with one of the statistics.

Spread
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Range

• Formula is high score low score.
• Example 4 1 5 3 3 6 1 2 6 4 5 3 4 1, N 14
• Arrange data in order 1 1 1 2 3 3 3 4 4 4 5 5 6
  6
• Range high score low score 6 1 5

range
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Midrange (MR)
midrange

• Formula is MRUH-LH.
• UHupper hinge
• LHlower hinge
• Hinges cut off 25 of the data in each tail
• Hinge position is (median position1)/2.
• median position is the whole number part of the
  median position (remember, median pos.(N1)/2)
• Use hinge position to count in from the tails to
  find the hinges.

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Midrange (MR)

• Example 4 1 5 3 3 6 1 2 6 4 5 3 4 1, N14
• Arrange data in order 1 1 1 2 3 3 3 4 4 4 5 5 6
  6
• Compute median position (N1)/2(141)/215/27
  .5
• Compute hinge position
• (median position1)/2(71)/28/24
• Count in to the 4th score from each tail to find
  UH and LH
• UH5 and LH2
• MRUH-LH5-23

midrange
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Sample Variance, s²
S

• Definitional formula s² ?(X-X)²
• N

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Sample Variance, s²
S

• Definitional formula s² ?(X-X)²
• N
• the average squared deviation from X.

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Sample Variance, s²
S

• Definitional formula s² ?(X-X)²
• N
• the average squared deviation from X.
• Example 1 2 3
• N3, X ?X/N6/32
• ?(X-X)² (1-2)²(2-2)²(3-2)²-1202121012
• s²2/N2/3.6667

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Sample Variance, s²
S

• Definitional formula s² ?(X-X)²
• N
• the average squared deviation from X.
• Example 1 2 3
• N3, X ?X/N6/32
• ?(X-X)² (1-2)²(2-2)²(3-2)²1012
• s²2/3.6667
• Computational formula s² N?X²-(?X)²
• N2
• ?X² 1²2²3²14914, ?X6, N3
• s²3(14)-(6)²/3²42-36/96/92/3.6667
• s² is in squared units of measure.

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Sample Standard Deviation, s

• Formula s ?s²
• Example 1 2 3
• N3, X ?X/N6/32
• ?(X-X)² (1-2)²(2-2)²(3-2)²1012
• s²2/3.6667
• s ?.6667.8165
• s is in original units of measure.

S
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Unbiased Variance Estimate, s²

• Definitional formula s² ?(X-X)²
• (N-1)
• Example 1 2 3
• N3, X ?X/N6/32
• ?(X-X)² (1-2)²(2-2)²(3-2)²1012
• s²2/21.0
• Computational formula
• s² N?X²-(?X)²
• N(N-1)
• ?X² 1²2²3²14914, ?X6, N3
• s²3(14)-(6)²/3(2)42-36/66/61.0
• s² is in squared units of measure

S
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Standard Deviation, s
S

• Formula s ?s²
• Example 1 2 3
• N3, X ?X/N6/32
• ?(X-X)² (1-2)²(2-2)²(3-2)²1012
• s²1.0
• s ?11.0
• s is in original units of measure.
• s is the typical distance of scores from the mean.

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Why do we care about measures of middle and
variability?

• Once weve collected data, the first step is
  usually to organize the information using simple
  descriptive statistics (e.g., measures of middle
  and variability)
• Measures of middle are AVERAGES. Mean, median,
  and mode are different ways of finding the one
  value that best represents all of your data
• Measures of variability tell us how scores DIFFER
  FROM ONE ANOTHER.

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What do those formulas mean?

• Computing the mean X ?X
  
  
• 
  N
• List the entire set of values in one or more
  columns. These are all the Xs.
• Compute the sum or total of the values.
• Divide the total or sum by the number of values.
• Computing the median (N1)/2 is the Median
  Position
• List the values in order (from lowest to
  highest).
• Find the middle-most score (i.e., the score in
  the median position). Thats the median.
• Computing the mode
• List the entire set of values, but list each only
  once.
• Tally the number of times that each value occurs.
• The value that occurs most often is the mode.

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What do those formulas mean?

• Computing the range Range Highest score
  Lowest score
• Find the highest and lowest scores.
• Subtract the lowest score from the highest.
• Computing the Midrange MR UH-LH
• Find the hinge position (median position1)/2.
  Use this to count in from the tails to find the
  hinges.
• Subtract the Lower Hinge from the Upper Hinge.
• Computing the Sample Variance (s2) ?(X-X)²
  
  
  N
• Compute the mean for the group.
• Subtract the mean from each score.
• Square each of these difference scores. (This
  gets rid of negative numbers).
• Sum all of the squared deviations around the
  mean.
• Divide the sum by N. (This gives you the AVERAGE
  SQUARED DEVIATION AROUND THE MEAN)

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Why do we have two formulas for variance and
standard deviation?

• Remember that our statistics are ESTIMATES of the
  parameters in the population.
• When we use N as the denominator (as in s2
  s), then we produce a biased estimate (it is too
  small).
• Since we are trying to be good scientists, we
  will be conservative and use the unbiased
  estimates of the variance and standard deviation
  (s2 s).
• Why did I confuse you with these different
  formulas? We will address the idea of bias
  later in the semester and this is a good
  introductionplus, some of the instructors for
  your later statistics courses may expect you to
  have a solid understanding of biased and unbiased
  estimates of variability.

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PRACTICE

• Data 1, 3, 5, 2, 13, 11, 1, 4
• Compute the mean
• X SX/N 40/8 5
• Compute the median
• Place scores in order 1, 1, 2, 3, 4, 5, 11, 13
• Median position (N1)/2 (81)/2 9/2 4.5
• Count in from the end 4.5 places
• X50 (34)/2 7/2 3.5
• Compute the mode
• Tally the scores
• Mode 1

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PRACTICE

• Data 1, 3, 5, 2, 13, 11, 1, 4
• Compute the Range
• Range High score Low score 13-1 12
• Compute the Midrange
• Place scores in order 1, 1, 2, 3, 4, 5, 11, 13
• MR UH - LH
• Hinge position (median position1)/2
  (41)/2 2.5
• Count in from each end 2.5 places
• UH (511)/2 8
• LH (12)/2 1.5
• MR UH LH 8 1.5 6.5

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PRACTICE

• Data 1, 3, 5, 2, 13, 11, 1, 4
• Compute s2
• s² ?(X-X)²
  
  N
• (1-5)2(3-5)2(5-5)2(2-5)2(13-5)2(11-5)2(1-5)
  2(4-5)2
  8
• (-4)2(-2)2(0)2(-3)2(8)2(6)2(-4)2(-1)2
  
  8
• 164096436161 146
  18.25 8 8
• Compute s
• s ?s²
• s ? 18.25 4.27

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PRACTICE

• Data 1, 3, 5, 2, 13, 11, 1, 4
• Compute s2
• s² ?(X-X)²
  
  
  N-1
• (1-5)2(3-5)2(5-5)2(2-5)2(13-5)2(11-5)2(1-5)
  2(4-5)2
  8-1
• (-4)2(-2)2(0)2(-3)2(8)2(6)2(-4)2(-1)2
  
  8-1
• 164096436161 146
  20.86 8-1
  7
• Compute s
• s ?s²
• s ? 20.86 4.57

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Homework for Next Time

• 8.8, 6.4, 8.4, 7.8, 8.8, 8.9, 7.8, 7.7, 4.5, 3.7,
  9.1, 8.3, 7.5, 3.2, 8.9, 4.4, 6.8
• Use SPSS to find N, Minimum score, Maximum score,
  Mean, Standard Deviation
• Which standard deviation does SPSS compute s or
  s?