# Quartiles, quintiles, deciles, and percentiles

### Merke

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{*LOSnr15*}
Describe, calculate, and interpret measures of central tendency. Include the following: population mean, sample mean, arithmetic mean, weighted average or mean (including a portfolio return viewed as a weighted mean), geometric mean, harmonic mean, median, and mode.

Quantiles divide a distribution into different parts, using the formula

Ly = (n + 1)∙(y/100).

y is the percentage point at which we divide the distribution. Ly is the location of the percentile in the array, sorted in ascending order.

### Beispiel

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For the numbers from example in "Frequency distributions"

, calculate the first quartile.

First we must arrange the numbers from that example in ascending order. So instead of writing

50, 48, 44, 32, 35, 60, 62, 55, 40, 45,

we should write

32, 35, 40, 44, 45, 48, 50, 55, 60, 62.

L25 = (10 + 1)∙(25/100) = 11∙0.25 = 2.75. It lies between the second and the third value, i.e. between 35 and 40. Here we have to use linear interpolation.

Interpolated value = X2 + (L25 – 2)*(X3 – X2)

= 35 + (2.75 – 2)*(40 – 35) = 35 + 0.75*5

= 38.75.

### Beispiel

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The median divides a distribution into halves, quartiles divide it into quarters, quintiles divide it into fifths, deciles divide it into tenths, and percentiles divide it into hundredths