1. Numerical Measures of Position

2. Standardization With the x & s
a clear picture of the distribution of a set of values is possible.
Often, an individual value is of interest rather than the entire set of values.

3. Standardization Using the x & s, a standard score can be calculated.
Standardization allows for
the comparison of individual scores,
even those with different scales
Ex. The ability to compare the results of 3 different writing measures
Ex. Weights of an individual calculated in lbs and kg

4. Standardization and z Score
Standardization is simply a transformation of a value to a linear standard score
most frequently a z score.
A z score tells us how far a raw value is above or below (±) the mean of a distribution (the group of values).
The distance is expressed using standard deviation units.
Standard Score
z = (x - x ) / s

5. z Score
z = (x - x ) / s
Ex. Ashley has two test scores on listening comprehension; Test A = 49 and Test B = 99.
Test A = 49 Test A has a M = 44 and a s = 4. What would her standard score be for Test A?
zA = ( ) / 4 = 5/4 = 1.25
Test B = 99 Test B has a M = 104 and a s = 3. What would her standard score be for Test B?
zB = ( ) / 3 = -5/3 = -1.67

6. z Scores and Normal Curve
z scores can be plotted on the Normal Curve.
There is a direct 1:1 ratio between standard deviation and z scores.
± 1s = ± 1z ± 2s = ± 2z ± 3s = ± 3z

7. z Scores and Normal Curve
z scores can be plotted on the normal curve
Test A = 1.25 z
Test B = z M S z

8. z Score Regardless of the scale, z scores remain the same
Patients’ z scores remain the same even though the measurement scale changes
Weight in kg Weight in lbs.
Student x z x z A B C D

9. Percentile Rank
Percentile ranks (%ile) are based on the percentage areas under the curve.
A percentile rank (PRx) describes the percentage of scores in the distribution that fall above and below the given standard score.
Ex.
A value at the 65th percentile falls at or above 65% of the values in the distribution.
A values at the 43rd percentile falls at or above 43% of the values in the distribution.

10. Percentile Rank The formula for Percentile Rank is as follows:
PRx = B fx = n
where B = number of values below x
where fx = number of values equal to x
With this measure, rank order is a must!!

11. Percentile Rank Shoe Sizes for a sample of 12 male students
What is the percentile rank for a shoe size of 12?
Rank Order
PR12 = (1) X 100 12
PR12 = 79.17
A value of 12 corresponds approx. to the 79th percentile.
79% of the sampled male students have a shoe size of 12 or smaller.
We round percentile ranks to the nearest whole number between 1 and 99.

12. Distributions Percent of values under portions of the normal curve
34.13%
34.13%
13.59%
13.59%
2.14%
2.14%
0.13%
0.13% S M

13. Percentile Rank and Normal Curve
Percentile Rank can be plotted on the normal curve
Size = 79th
Size = 50th
Size = 67th M S z
%ile S z