Measures of Position (College Board AP® Statistics)

Recall that the median splits the data set into two parts

• At least 50% of the data is less than or equal to the median

• and 50% of the data is greater than or equal to it

Quartiles split the data set into four parts

• The first quartile (Q1) lies a quarter of the way along the data (when in order)

  • One quarter (25%) of the data is less than or equal to Q1 (and three quarters is greater than or equal to it)

• The third quartile (Q3) lies three quarters of the way along the data (when in order)

  • Three quarters (75%) of the data is less than or equal to Q3 (and one quarter is greater than or equal to it)

• You may come across the median being referred to as the second quartile (Q2)

Make sure the data is written in numerical order

Use the median to divide the data set into lower and upper halves

• If there are an even number of data values, then

  • the first half of those values are the lower half,

  • and the second half are the upper half

  • All of the data values are included in one or the other of the two halves

• If there are an odd number of data values, then

  • all the values below the median are the lower half

  • and all the values above the median are the upper half

  • The median itself is not included as a part of either half

The first quartile is the median of the lower half of the data set

• and the third quartile is the median of the upper half of the data set

Find the quartiles in the same way you would find the median for any other data set

• just restrict your attention to the lower or upper half of the data accordingly

A percentile indicates the position of a data value within its distribution

Percentiles divide a data set into 100 equal parts

• % of the data values will be less than or equal to the ^th percentile

  • e.g. 10% of data values will be less than or equal to the 10^th percentile (and 90% will be greater than or equal to it)

  • 99% of data values will be less than or equal to the 99^th percentile (and 1% will be greater than or equal to it)

• Note that

  • the 25^th percentile is the same as the first quartile

  • the 50^th percentile is the same as the median

  • the 75^th percentile is the same as the third quartile

• Also note that percentiles don't need to be whole numbers

  • e.g. we can talk about the 2.5^th percentile

  • 2.5% of the data will be less than or equal to that (and 97.5% will be greater than or equal to it)

Percentiles can be useful for discussing the distribution of data in a data set

• e.g. in comparing incomes in the US

  • you could compare the highest 1% of earners (the ones above the 99^th percentile)

  • with the median income (the 50^th percentile)

  • or the lowest 10% of earners (those below the 10^th percentile)

To find a percentile

• Write the percentile as a fraction or a decimal

  • e.g. 45^th percentile is or 0.45

• Determine the total number of items in the data set

• Multiply the total number of items by the percentile

• This will give you the position in the data set (when ordered) of the percentile required

You may be required to find a single value or a class interval in which the percentile is situated

• this could apply to data in a table

• or in a diagram, e.g. a stem-and-leaf plot or a dotplot