The Normal Distribution and Measures of Relative Standing

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The Normal Distribution and Measures of Relative
Standing
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Z-Scores or Standard Scores

• Often, we need a measure to express where a score
  stands in relation to the entire group. Such a
  score is a measure of relative standing.
• The Z-score or standard score is such a measure.
  It tells us the number of standard deviations a
  single score is from the mean.

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• The sample z-score is calculated by subtracting
  the sample mean from the individual raw score and
  then dividing by the sample standard deviation.

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• The population z-score is calculated by
  subtracting the population mean from the
  individual raw score and then dividing by the
  population standard deviation.

Where x is the individual score µ
is the population mean ? is the population
standard deviation
Z X - µ ?
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E.g., Suppose 40 people were administered a
cognitive test where the mean score was 55 and
the standard deviation was 5. Find the z-scores
associated with the following 4 participants
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Z-Scores and the Normal Distribution

• If we have a normal distribution we can make the
  following assumptions.
• Approximately 68 of the scores are between a
  z-score of 1 and -1.
• Approximately 95 of the scores will be between a
  z-score of 2 and -2.
• Approximately 99.7 of the scores will be between
  a z-score of 3 and -3.

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Z-Scores and Percentile Ranks

• Z-scores may be used to calculate the other
  measure of relative standing, i.e., percentile
  ranks.
• When used in conjunction with Table A (page 325),
  the z-score reveals the area of the normal
  distribution between the score in question and
  the mean.

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• The z-score for x gives the area from x to the
  mean. This represents the percentage of those in
  the data set that score between x and the mean.
  To get percentile for x, we add this to 0.5 from
  the first part of the distribution

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An Example

• On a recent stats exam, the class average was 65
  with a standard deviation of 10.
• Dave and Tom scored 82 and 70 respectively, what
  are their percentile ranks?
• Step 1 Calculate the z-scores.

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Dave z 1.7

• Step 2 Plot these points on the distribution.

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• Step 3 Use Table A to find the percentage from
  the score to the mean.
• Step 4 Add the total from step 3 to the area
  below the mean, i.e., 0.5. Then multiply by 100.
• (0.4554 0.5) x 100 95.54
• Dave scored in the 96th percentile.

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Tom z 0.5

• Step 2 Plot these points on the distribution.

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• Step 3 Use Table A to find the percentage from
  the score to the mean.
• Step 4 Add the total from step 3 to the area
  below the mean, i.e., 0.5. Then multiply by 100.
• (0.1915 0.5) x 100 69.15
• Tom scored in the 69th percentile.

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Another Example

• Calculating percentile ranks for scores below the
  mean is slightly more difficult.
• Steve and Mark were in the same stats class as
  Dave and Tom (x 65, s 10).
• Steve scored 60 and Mark scored 51.
• What were their percentile ranks?

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Because we know the lower half of the curve
has an area of 0.5, we can take 0.5 and
subtract 0.1915 to find the area below a score
of 60, i.e., we can find the percentile
rank. - so (0.5 - 0.1915) x 100 30.85
Steve scored in the 31st percentile.
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Because we know the lower half of the curve
has an area of 0.5, we can take 0.5 and
subtract 0.4192 to find the area below a score
of 51, i.e., we can find the percentile
rank. - so (0.5 - 0.4192) x 100 8.08 Mark
scored in the 8th percentile.
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Another Example

• We can also find the percentage of students who
  score between Mark (51) and Dave (82).

We simply add the areas we obtained from Marks
and Daves z- scores. (0.4192 0.4554) x 100
87.46 Approximately 87.5 of the students scored
between Mark and Dave.
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• If we know the total number of students in the
  class, we can calculate the the number of
  students who finished between Mark and Dave.
• Multiply total N by the area under the curve
  between Mark and Dave
• E.g., Assume there were 60 people in the class.

No. between Mark and Dave 60 x 0.875
53
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Working Backwards

• We can also use the z-score to work backwards and
  figure out x.
• E.g., Kim is in the same stats class as Steve,
  Mark, Dave and Tom. She finished in the 65th
  percentile. What was her grade?
• X 65, S 10

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• Step 1 - Draw the distribution and figure out the
  area between x and the mean.

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• Step 2 - Look up the area from Step 1 in Table A
  (pp. 325-327) to figure out the z-score.
• The closest to 0.15 is .1517.
• This corresponds to a z-score of 0.39.

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• Step 3 - Plug this z-score into the z formula and
  solve for X.

Z X - X S
0.39 X - 65 10
10(0.39) 65 X
X 68.9
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Another Example

• Two hundred students wrote a final exam for a
  first year psych class. The mean of the exam was
  60 and the standard deviation was 12.
• If Mike scored in the 75th percentile, what was
  his score?

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• Find the z-score that corresponds to an area of
  0.25.

The closest is Z 0.67
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• Solve for X.

0.67 X - 60 12
Z X - X S
X 68.04
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Another Example

• Kate also wrote the psych exam and got an 85.
  How many people scored between Kate and Mike?

Z 85 - 60 12
Z 2.08
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We know that Kate scored in the 98th percentile,
but we need to know how many students finished
between her and Mike.
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Area between 68.04 85 0.4812 - 0.25 0.2312
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• Remember we were asked for the number of students
  who finished between Kate and Mike.

Number between 68.04 85 200 x 0.2312
46.24
46 students finished between Mike and Kate.
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Another Example

• The average weight of a sample of 200 males at
  MUN is 165 lbs with a standard deviation of 25.
• Mike, and Bill weigh 175 and 200 lbs
  respectively.
• Calculate the percentile rank of each.

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Mike z 0.4
(0.5 0.1554) x 100 65.54 66th percentile
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Bill z 1.4
0.5
0.4192
165
200
(0.5 0.4192) x 100 91.92 92nd percentile
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• What score is at the 30th percentile?

0.20

• Find the z-score that corresponds to an area of
  0.20.

The closest is Z -0.52
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• Solve for X.

-0.52 X - 165 25
Z X - X S
X 152
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• Between what two scores does the middle 30 of
  the sample lie?

0.15
0.15
165
X
X

• Find the z-scores that correspond to an area of
  0.15.

The closest is Z 0.39
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• Solve for X.

-0.39 X - 165 25
X 155.25
0.39 X - 165 25
X 174.75
The middle 30 of the subjects lie between 155.25
and 174.75 lbs.