12.4 – Measures of Position

It is necessary at times, to be able to measure how an item fits into the data, how it compares to other items of the data, or even how it compares to another item in another data set.

In some cases, the analysis of certain individual items in the data set is of more interest rather than the entire set.

Measures of position are several common ways of creating such comparisons.

The z-Score

.x x

zs

12.4 – Measures of Position

The z-score measures how many standard deviations a single data item is from the mean.

Example: Comparing with z-Scores

Two students, who take different history classes, had exams on the same day. Jen’s score was 83 while Joy’s score was 78. Which student did relatively better, given the class data shown below?

Jen Joy

Class mean 78 70

Class standard deviation 4 5

12.4 – Measures of Position

Example: Comparing with z-Scores

Jen 83

Joy 78

Class mean 78 70

Class standard deviation 4 5

12.4 – Measures of Position

Jen’s z-score:

83 – 78

4= 1.25

Joy’s z-score:

78 – 70

5= 1.6

Joy’s z-score is higher as she was positioned relatively higher within her class than Jen was within her class.

Standardized tests taken by larger numbers of students, convert raw scores to a percentile score.

Percentiles

12.4 – Measures of Position

If approximately n percent of the items in a distribution are less than the number x, then x is the nth percentile of the distribution, denoted Pn.

A percentile measure the position of a single data item based on the percentage of data items below that single data item.

Example:

The following are test scores (out of 100) for a particular math class.44 56 58 62 64 64 70 72 72 72 74 74 75 78 78 79 80 82 82 84 86 87 88 90 92 95 96 96 98 100

Find the fortieth percentile.

Percentiles

12.4 – Measures of Position

40% = 0.4

0.4(30)

12 40% of the scores were below 74.5.

The average of the 12th and 13th items represents the 40th percentile (P40).

Deciles are the nine values (denoted D1, D2,…, D9) along the scale that divide a data set into ten (approximately) equal parts.

Other Percentiles: Deciles and Quartiles

12.4 – Measures of Position

Quartiles are the three values (Q1, Q2, Q3) that divide the data set into four (approximately) equal parts.

10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, and 90%

25%, 50%, and 75%

Example: DecilesThe following are test scores (out of 100) for a particular math class.44 56 58 62 64 64 70 72 72 72 74 74 75 78 78 79 80 82 82 84 86 87 88 90 92 95 96 96 98 100

Find the sixth decile.

Other Percentiles: Deciles and Quartiles

12.4 – Measures of Position

Sixth decile = 60%

60% = 0.6

0.6(30)

1860% of the scores were at or below 82.

The average of the 18th and 19th items represents the 6th decile (D6).

Quartiles

For any set of data (ranked in order from least to greatest):

Other Percentiles: Deciles and Quartiles

12.4 – Measures of Position

The first quartile, Q1 (25%) is the median of items below Q2.

The second quartile, Q2 (50%) is the median.

The third quartile, Q3 (75%) is the median of items above Q2.

Example: QuartilesThe following are test scores (out of 100) for a particular math class.44 56 58 62 64 64 70 72 72 72 74 74 75 78 78 79 80 82 82 84 86 87 88 90 92 95 96 96 98 100

Find the three quartiles.

Other Percentiles: Deciles and Quartiles

12.4 – Measures of Position

Q1= 25%

25% = 0.25

0.25(30)

7.5

The 8th item represents the 1st quartile (Q1)

25% of the scores were below 72.

Example: QuartilesThe following are test scores (out of 100) for a particular math class.44 56 58 62 64 64 70 72 72 72 74 74 75 78 78 79 80 82 82 84 86 87 88 90 92 95 96 96 98 100

Find the three quartiles.

Other Percentiles: Deciles and Quartiles

12.4 – Measures of Position

Q2= 50% = median

50% = 0.5

0.5(30)

15

The average of the 15th and 16th items represents the 2nd quartile (Q2) or the median50% of the scores were below 78.5.

Example: QuartilesThe following are test scores (out of 100) for a particular math class.44 56 58 62 64 64 70 72 72 72 74 74 75 78 78 79 80 82 82 84 86 87 88 90 92 95 96 96 98 100

Find the three quartiles.

Other Percentiles: Deciles and Quartiles

12.4 – Measures of Position

Q3= 75%

75% = 0.75

0.75(30)

22.5

The 23rd item represents the 3rd quartile (Q3)

75% of the scores were below 88.

A box plot or a box and whisker plot is a visual display of five statistical measures.

Box Plots

12.4 – Measures of Position

The five statistical measures are:

the lowest value,

the first quartile, the median, the third quartile,

the largest value.

the lowest value

the largest value

Box Plots

12.4 – Measures of Position

Example:The following are test scores (out of 100) for a particular math class.44 56 58 62 64 64 70 72 72 72 74 74 75 78 78 79 80 82 82 84 86 87 88 90 92 95 96 96 98 100

Q3= 75%= 88Q2= 50% = median= 78.5Q1= 25% = 72

Lowest = 44 Largest = 100