Standard Score

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Transcript Standard Score

Unit 3
Section 3-4
3-4: Measures of Position
 Measures
of position are used to locate
the position of a data value within a data
set.
 Measures of Position
 Standard
Score (z score)
 Percentiles
 Deciles
 Quartiles
Section 3-4
 Standard
Score – a value obtained by
subtracting the mean from the value and
dividing the result by the standard
deviation.
 Also
known as a z score
 Symbol: z
 Represents the number of standard deviations
that a data value falls above or below the
mean.

Positive values are above the mean, negative are
below the mean.
Section 3-4
Standard Scores

A student scored a 65 on a calculus test that
had a mean of 50 and a standard deviation
of 10; she scored 30 on a history test with a
mean of 25 and a standard deviation of 5.
Compare her relative position on the two
tests.



Find the standard score for calculus
Find the standard score for history
The larger the z score, the higher her position in
that class.
Section 3-4
Standard Scores
 Find
the z score for each test. State which
is higher.
Test
Value
Mean
Standard
Deviation
Test A
38
40
5
Test B
94
100
10
Section 3-4
 Percentiles–
divide the data set into 100
equal groups.
 Ranks
a data value based on its position
within the data set.
 Example: a percentile of 80% means that the
specific data value is higher than 80% of the
values within the data set.
 Percentiles often used to rank performance
on individual tests such as PSAT
Section 3-4
Finding the Percentile
 Arrange
the data in order from lowest to
highest.
 Locate the corresponding data value.
 Count the number of values smaller than
the corresponding value.
 Take that value, add 0.5
 Divide by the total number of values.
 Multiply by 100
 The result is the data value’s position as a
percent.
Section 3-4
Finding a Percentile
A
teacher gives a 20-point test to 10
students. The scores are shown below.
18, 15, 12, 6, 8, 2, 3, 5, 20, 10
a) Find the percentile rank of a score of 12.
b) Find the value corresponding to the 25th
percentile.
Section 3-4
 Quartiles–
groups.
divide the data set into 4 equal
 The
quartiles are separated by the values Q1,
Q2, and Q3.
 Q1 - 25th percentile
 Q2 – 50th percentile, or the median
 Q3 – 75th percentile
 Interquartile
Range (IQR) – the difference
between Q3 and Q1.
Section 3-4
Finding the Quartile
 Arrange
the data in order from lowest to
highest.
 Find the median of the data. (Q2)
 Find the median of the first half of the
data. (Q1)
 Find the median of the second half of the
data. (Q3)
Section 3-4
Finding a Quartile
 Find
Q1, Q2, and Q3 for the data set:
15, 13, 6, 5, 12, 50, 22, 18
Section 3-4
 Deciles–
groups.
divide the data set into 10 equal
 The
deciles are labels as D1, D2, D3, … , D9.
 D1 - 10th percentile
 D2 – 20th percentile
 D3 – 30th percentile …
 Outlier
– an extremely high or an
extremely low data value when
compared with the rest of the data
values.
Section 3-4
Finding an Outlier
 Arrange
the data in order to find Q1 and
Q3.
 Find the IQR (Q3 – Q1).
 Multiply the IQR by 1.5


Lower Boundary: subtract that value from
Q1
Upper Boundary: add that value to Q3.
 Check
the data set for any value that is
outside the boundaries.
Section 3-4
Finding the Outlier
 Check
the following data set for outliers.
5, 6, 12, 13, 15, 18, 22, 50
Section 3-4
Homework
 Pg
144: 1-9, 13