4.2.4 Percentiles of a Continuous Distribution

• When we say that an individual’s test score was at the 85th percentile of the population, we mean that
  • $85\%$ of all population scores were below that score and $15\%$ were above.
• Similarly, the 40th percentile is the score that
  • exceeds $40\%$ of all scores and is exceeded by $60\%$ of all scores
  • having a value corresponding to a high percentile is not necessarily good;
    • e.g., you would not want to be at the 99th percentile for blood alcohol content

percentile

Let $p$ be a number between 0 and 1. The $\left(100p\right)$ th percentile of the distribution of a continuous rv $X$, denoted by $\eta \left(p\right)$, is defined by

$$p=F\left(\eta \left(p\right)\right)={\int }_{-\infty }^{\eta \left(p\right)}f\left(y\right)dy\text{\hspace{1em}(4.2)}$$

According to Expression (4.2), $\eta \left(p\right)$ is that value on the measurement axis such that

• $100p\%$ of the area under the graph of $f\left(x\right)$ lies to the left of $\eta \left(p\right)$
• $100\left(1-p\right)\%$ lies to the right.

Thus $\eta \left(.75\right)$, the 75 th percentile, is such that

• the area under the graph of $f\left(x\right)$ to the left of $\eta \left(.75\right)$ is .75.

Figure 4.10 illustrates the definition.

Figure 4.10 The $\left(100p\right)$ th percentile of a continuous distribution

EX 4.9 amount of gravel

median

The median of a continuous distribution, denoted by $\tilde{\mu }$ , is the 50th percentile, so $\tilde{\mu }$ satisfies $.5=F\left(\tilde{\mu }\right).$ That is, half the area under the density curve is to the left of $\tilde{\mu }$ and half is to the right of $\tilde{\mu }$ .

A continuous distribution whose pdf is symmetric

• the graph of the pdf to the left of some point is a mirror image of the graph to the right of that point-has median $\tilde{\mu }$ equal to the point of symmetry, since half the area under the curve lies to either side of this point.

Figure 4.12 gives several examples. The error in a measurement of a physical quantity is often assumed to have a symmetric distribution.

Figure 4.12 Medians of symmetric distributions