PEP 6305 Measurement in Health & Physical Education

Topic 3: Percentiles & Measures of Central Tendency

Section 3.1

n   This Topic has 2 Sections.

Readings

n   Vincent & Weir, Statistics in Kinesiology 4th ed., Chapter 3: “Percentiles” and Chapter 4: “Measures of Central Tendency”

Purpose

n   To demonstrate the computation of percentiles for a distribution of scores.

n   To discuss and demonstrate various measures of the central tendency of a distribution of scores.

Percentiles

n   A percentile is a point on a scale where a certain proportion of the population lies at or below.

¨  Example: a score at the 75^th percentile means that 75% of the population have scores at or below that value.

n   Percentiles are a type of standard score, which means that a raw score is converted into a score that has a known (“standard”) meaning.

¨  Usually the standard is a comparison to the distribution of scores in a population (of subjects or other objects being compared) or a comparison to a known range of values (as is the case in laboratory measures).

n   In the case of percentiles, the known range is 0% to 100% and the midpoint (with half of the population above and half below) is 50%.

n   Percentiles can be used to determine where a person’s performance or rating lies in relation to the population.

Common Percentile Divisions

n   Because measurement can never be perfect, ranges of scores and ranges of percentiles are often used to report test results.

¨  Since a person's score varies slightly from test to test, these intervals often provide a more fair representation of a person’s abilities than an exact percentile score.

¨  Two examples of these ranges of percentiles are quartiles and deciles.

n   Quartiles (four divisions, ranges of 25 percentile units): quartile 1 = 0 to 25^th percentile (Q1), quartile 2 = top of Q1 to 50^th percentile (Q2), quartile 3 = top of Q2 to 75^th percentile (Q3), quartile 4 = top of Q3 to 100^th percentile. Each quartile contains one-fourth (25%) of the scores in the distribution. See Figure 3.2 in the text.

n   Deciles (ten divisions, D1 to D10): similar to quartiles, but progressing by ranges of 10 percentile units; decile 1 = 0 to 10^th percentile (D1); decile 2 = top of D1 to 20^th percentile, etc. See Figure 3.2 in the text.

n   Remember: a quartile or decile is a range of percentiles; a specific percentile is equivalent to a single score (in standardized form).

n   A quartile rank or decile rank means that the person’s score lies within a certain quartile or decile.

¨  Example: someone with a decile rank of 9 means that their score lies in the 9^th decile—their score is between the 80^th and 90^th percentile.

Calculating Percentiles

n   The ways to calculate percentiles vary depending on whether you have all of the data or only have summaries such as a simple frequency distribution or a grouped frequency distribution.

n   If you have all of the data, create a rank order distribution. Using the rank order distribution, for each score, figure out how many scores fall at or below that score. Divide that number by the total number of scores. Multiply by 100 and round to the nearest whole percent value. That percent value is the percentile.

¨  Example: In Topic 2, you created a rank order distribution for these data: 525, 505, 507, 654, 631, 281, 771, 575, 485, 626, 780, 626. What are the percentiles for each value in this distribution? (Click to see an Excel file solution to this problem.)

n   If you have only summaries of the data, use Equation 3.01 from the text:

where X is the score, L is the lower real limit of the interval (see the text, p. 26), i is the width of the interval, f is the frequency in the interval, C is the cumulative frequency of the next lowest interval, and N is the total number of subjects/cases.

¨  Example: for the grouped frequency distribution in Table 3.3 in the text, the percentile for a softball throw of 195 feet is:

     , or approximately the 85^th percentile.

¨  For the score of 195 in this distribution, X = 195; L = 189.5; i = 10; f = 7; C = 94; and N = 115.

¨  What are the percentiles for softball throws of 200 feet, 175 feet, 150 feet, and 100 feet? (Click HERE for answers.)

n   It is also possible to estimate a score for a given percentile (if you have the simple frequency distribution for the scores).

¨  Convert the percentile to a decimal. For example, the decimal for a percentile of 75 is 0.75; for 50 is 0.50; and for 10 is 0.10.

¨  Multiply that decimal value by the number of scores in the set of data being analyzed. For example, if you want the 50th percentile for 40 scores: 0.50 × 40 = 20.

¨  Count that number of scores in a simple frequency distribution, starting from the bottom (lowest/worst score) up to that number of scores.

¨  For the example, count from the lowest score to the 20th score. That score (in this case, 20th from the bottom) is the raw score for the respective percentile.

¨  If the number of scores to count is not an integer, round up to the next whole number and use that score. For example, if you want the 23rd percentile for 40 scores: 0.23 × 40 = 9.2. In this case you count to the 10th score from the bottom, and that is your raw score for the 23rd percentile.

n   Click here to see solutions worked in Excel to Chapter 3 Problems 1 and 2 (compute percentiles from scores and vice-versa). (Please work them yourself, then check your work against this key.)

n   To obtain decile values in R Commander, load your data file and go to the Statistics menu, Summaries>Numerical summaries...

¨  In the dialog box, click to highlight age to the Variables box and enter the numbers 0, .1, .2, .3, .4, .5, .6, .7, .8, .9, 1 in the 'quantiles" box.

     ¨  Click OK. A table showing the decile values for age appears in the Output window: