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  Theory of Measures of Position

Measures of Position:

Standard Scores (z-scores):

It can be obtained by subtracting the mean and dividing the difference by the standard deviation. The symbol is z that is why it is as well termed as z-score.

z = (x - μ)/σ or z = (x –x¯)/s

For mean, the standard scores is zero and for standard deviation it is 1. This is the nice characteristic of the standard score – the original scale was not the matter, when the data is transformed to its standard score, the mean is zero and standard deviation is 1.

Percentiles, Deciles and Quartiles:

Percentiles (100 regions):

The kth percentile is the number that has k% of the values under it. The data should be ranked.

•    At first, rank the data
•    Determine k% (k/100) of the sample size, n.
•    When this is an integer, add 0.5. When it isn't an integer, then round up.
•    Determine the number in this place. When your depth ends in 0.5, then take the midpoint among the two numbers.

Counting from the high end instead of counting from the low end is sometimes easy. For illustration, the 80th percentile is the number that has 80% beneath it and 20% above it. Instead of counting 80% from the bottom, count 20% from top.

Note: The 50th percentile is the median.

For evaluating the percentile for a number instead of locating the kth percentile, follow the steps:

i) Take the number of values beneath the number.
ii) Add 0.5.
iii) Divide by total number of values.
iv) Transform it to a percent.

Deciles (10 regions):

We know that the percentiles divide the data into 100 equivalent regions. In deciles the data is divided into 10 equivalent regions. The instructions are similar for finding a percentile, except rather of dividing by 100 in step 2, divide by 10.

Quartiles (4 regions):

In this, the quartiles divided the data into 4 equivalent regions. Rather than dividing by 100 in step 2, divide it by 4.

Note: The 2nd quartile is similar as the median. The 1st quartile is the 25th percentile; and the 3rd quartile is the 75th percentile.

The quartiles are generally used (much more than the percentiles or deciles). The TI-82 calculator will determine the quartiles for you.

Hinges:

Lower hinge can be stated as the median of lower half of the data up to and comprising the median. Upper hinge can be stated as the median of upper half of the data up to and comprising the median.

The hinges are similar as the quartiles except the remainder if dividing the sample size by four is three (similar to 39/4 = 9 R 3).

The statement regarding the lower half or upper half comprising the median tends to be confusing to various students. When the median is splitted between two values (that occurs if the sample size is even), the median isn't involved in either as the median isn't actually a part of data.

Illustration: Sample size of 20

The median will be in place of 10.5. Lower half positions at 1 to 10 and the upper half is at 11 to 20. The lower hinge is the median of lower half and would be in place 5.5. The upper hinge is the median of upper half and would be in place 5.5 beginning with original place 11 as place 1 -- the original place is 15.5.

Illustration: sample size of 21

The median is in place 11. The lower half is at position 1 to 11 and the upper half is at position 11 to21. The lower hinge is the median of lower half and would be in place 6. The upper hinge is the median of upper half and would be in place 6 whenever beginning at position 11 -- the original place is16.

Five Number Summary:

The five number summary comprises of the minimum value, median, lower hinge, upper hinge and maximum value. Some of the textbooks use the quartiles rather than hinges.

Box and Whiskers Plot:

It is a graphical representation of five number summary. The box is drawn between the lower and upper hinges with a line at median. Whiskers (that is, a single line, not a box) expand from the hinges to lines at minimum and maximum values.

Interquartile Range (IQR):

The difference between the third and first quartiles is usually termed as Interquartile Range. That's it: Q3 - Q1

Outliers:

Outliers are the extreme values. There are two kind of outliers are present: Mild outliers and Extreme outliers. The Bluman text doesn’t differentiate between mild outliers and extreme outliers and merely treats either as an outlier.

Extreme Outliers:

Extreme outliers are the data values that lay more than 3.0 times the interquartile range beneath the first quartile or above the third quartile. x is an extreme outlier when:

x < Q1 - 3 * IQR or
x > Q3 + 3 * IQR

Mild Outliers:

Mild outliers are the data values that lay between 1.5 times and 3.0 times the interquartile range beneath the first quartile or above the third quartile. x is a mild outlier when:

Q1 - 3 * IQR <= x < Q1 - 1.5 * IQR or
Q1 + 1.5 * IQR < x <= Q3 + 3 * IQR

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