WorldClass Statistics · Probability

Continuous distributions

far in this chapter we’ve discussed cases where the outcome of a variable is discrete. In this section, we consider a context where the outcome is a continuous numerical variable.

Example 3.71

Figure 3.24 shows a few different hollow histograms for the heights of US adults. How does changing the number of bins allow you to make different interpretations of the data?

Adding more bins provides greater detail. This sample is extremely large, which is why much smaller bins still work well. Usually we do not use so many bins with smaller sample sizes since small counts per bin mean the bin heights are very volatile.

Four hollow histograms are shown for the US adult heights in centimeters with varying bin widths. The difference in appears will first be discussed, and then the shape of the last, most detailed histogram will be given. The first histogram has about 6 bins with values that appear to be non-zero, so the outline is very boxy. The second histogram has about 12 non-zero bins, and appears a bit more continuous and less boxy than the first histogram. The third histogram has about 25 non-zero bins, and the hollow histogram outline is starting to look somewhat smoother. The last histogram has about 50 bins, and due to the large number of bins, the distribution looks quite smooth, in that no steps from one bin to the next is a substantial jump or drop in height. Next, this last histogram is described: The bin heights are about zero until 147, then they steadily climb up to about 155 before steeply climbing a little until 157 and then steadily climb to a peak at about 165. From here the histogram declines about 10% from its peak at 170, at which point the decline is more gradual until about 183, at which point it descends rapidly until about 187 where it begins to descend more slowly as it approaches 0. At about 200, the bin heights have essentially hit zero and stay there.

Figure 3.24. Four hollow histograms of US adults heights with varying bin widths.

Example 3.72

What proportion of the sample is between 180 cm and 185 cm tall (about 5’11" to 6’1")?

We can add up the heights of the bins in the range 180 cm and 185 and divide by the sample size. For instance, this can be done with the two shaded bins shown in Figure 3.25. The two bins in this region have counts of 195,307 and 156,239 people, resulting in the following estimate of the probability: 195307+1562393,000,000=0.1172\begin{aligned} \frac{195307 + 156239}{\text{3,000,000}} = 0.1172 \end{aligned} This fraction is the same as the proportion of the histogram’s area that falls in the range 180 to 185 cm.

A histogram for heights is shown, with the two histogram bins between 180 and 185 centimeters are shaded, representing the individuals with heights between 180 and 185 centimeters.

Figure 3.25. A histogram with bin sizes of 2.5 cm. The shaded region represents individuals with heights between 180 and 185 cm.

From histograms to continuous distributions

Examine the transition from a boxy hollow histogram in the top-left of Figure 3.24 to the much smoother plot in the lower-right. In this last plot, the bins are so slim that the hollow histogram is starting to resemble a smooth curve. This suggests the population height as a continuous numerical variable might best be explained by a curve that represents the outline of extremely slim bins.

This smooth curve represents a probability density function (also called a density or distribution), and such a curve is shown in Figure 3.26 overlaid on a histogram of the sample. A density has a special property: the total area under the density’s curve is 1.

A histogram for heights of US adults is shown with an overlaid continuous line along the heights of the bins. This continuous line is smooth and would represent what a hollow histogram would look like if we had infinite data and the bin width was so small that the boxy outline of the histogram looks continuous. This is called a "continuous probability density".

Figure 3.26. The continuous probability distribution of heights for US adults.

Probabilities from continuous distributions

We computed the proportion of individuals with heights 180 to 185 cm in Example 3.72 as a fraction: number of people between 180 and 185total sample size\begin{aligned} \frac{\text{number of people between \texttt{180} and \texttt{185}}}{\text{total sample size}} \end{aligned} We found the number of people with heights between 180 and 185 cm by determining the fraction of the histogram’s area in this region. Similarly, we can use the area in the shaded region under the curve to find a probability (with the help of a computer): P(height between 180 and 185)=area between 180 and 185=0.1157\begin{aligned} P(\text{\texttt{height} between \texttt{180} and \texttt{185}}) = \text{area between \texttt{180} and \texttt{185}} = 0.1157 \end{aligned} The probability that a randomly selected person is between 180 and 185 cm is 0.1157. This is very close to the estimate from Example 3.72: 0.1172.

A density curve for heights is shown, with the region between 180 and 185 centimeters being shaded.

Figure 3.27. Density for heights in the US adult population with the area between 180 and 185 cm shaded. Compare this plot with Figure 3.25.

Guided Practice 3.73

Three US adults are randomly selected. The probability a single adult is between 180 and 185 cm is 0.1157.1

  1. What is the probability that all three are between 180 and 185 cm tall?

  2. What is the probability that none are between 180 and 185 cm?

Example 3.74

What is the probability that a randomly selected person is exactly 180 cm? Assume you can measure perfectly.

This probability is zero. A person might be close to 180 cm, but not exactly 180 cm tall. This also makes sense with the definition of probability as area; there is no area captured between 180 cm and 180 cm.

Guided Practice 3.75

Suppose a person’s height is rounded to the nearest centimeter. Is there a chance that a random person’s measured height will be 180 cm?2

Exercises

Exercise 3.37

Cat weights. The histogram shown below represents the weights (in kg) of 47 female and 97 male cats. 3

  1. What fraction of these cats weigh less than 2.5 kg?

  2. What fraction of these cats weigh between 2.5 and 2.75 kg?

  3. What fraction of these cats weigh between 2.75 and 3.5 kg?

 \:

A histogram of cat body weights in kilograms is shown. The weight range is from 2.0 to 4.0, and each histogram bin has a width of 0.25. The eight bin heights, from left to right, are 29, 32, 21, 25, 12, 15, 5, and 4.

Exercise 3.38

Income and gender. The relative frequency table below displays the distribution of annual total personal income (in 2009 inflation-adjusted dollars) for a representative sample of 96,420,486 Americans. These data come from the American Community Survey for 2005-2009. This sample is comprised of 59% males and 41% females. 4

  1. Describe the distribution of total personal income.

  2. What is the probability that a randomly chosen US resident makes less than $50,000 per year?

  3. What is the probability that a randomly chosen US resident makes less than $50,000 per year and is female? Note any assumptions you make.

  4. The same data source indicates that 71.8% of females make less than $50,000 per year. Use this value to determine whether or not the assumption you made in part (c) is valid.

Income Total
$1 to $9,999 or loss 2.2%
$10,000 to $14,999 4.7%
$15,000 to $24,999 15.8%
$25,000 to $34,999 18.3%
$35,000 to $49,999 21.2%
$50,000 to $64,999 13.9%
$65,000 to $74,999 5.8%
$75,000 to $99,999 8.4%
$100,000 or more 9.7%

Notes

  1. Brief answers: (a) 0.1157×0.1157×0.1157=0.00150.1157 \times 0.1157 \times 0.1157 = 0.0015. (b) (1−0.1157)3=0.692(1-0.1157)^3 = 0.692 ↩

  2. This has positive probability. Anyone between 179.5 cm and 180.5 cm will have a measured height of 180 cm. This is probably a more realistic scenario to encounter in practice versus Example 3.74. ↩

  3. W. N. Venables and B. D. Ripley. Modern Applied Statistics with S. Springer. www.stats.ox.ac.uk/pub/MASS4. 2002. ↩

  4. U.S. Census Bureau, 2005-2009 American Community Survey. ↩

Credit and licence

“3.5 Continuous distributions” is from OpenIntro Statistics, Fourth Edition, by David M Diez, Mine Çetinkaya-Rundel and Christopher D Barr, shared under a CC BY-SA 3.0 license. WorldClass Statistics is a derivative of OpenIntro Statistics, shared under the same license; the original book may be found at openintro.org/book/os.

Source: https://www.openintro.org/book/os/

Licence: CC BY-SA 3.0 (creativecommons.org/licenses/by-sa/3.0/). You may share and adapt it under the same licence, with this credit. WorldClass adapted the formatting only: the text is the authors', with interactive parts and links to other sites left out.

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