Discrete to Continuous

The probability that a person picked at random has a height between 150 and 160 cm can be calculated from the table:

\[p=\frac{41}{15+41+67+72+65+34+3}\]

You could also find the probability by looking at the areas of the bars in bar chart. The probability is then found by dividing the area of the red bar by the total area.

\[p=\frac{41\cdot 10}{15\cdot 10+41\cdot 10+67\cdot 10+72\cdot 10+65\cdot 10+34\cdot 10+3\cdot 10}\]

Probability and area

One way of illustrating the probability that an outcome is in a certain interval, is to let the area of the interval be the probability. The total area must in this case be one. A new bar chart having the same appearance but the total area of one can be constructed.

Exercise 1

Add another column in the spreadsheet representing the relative frequency.

If you make a bar chart using the relative frequency instead of the absolute frequency, the total area will still not be one. Add another column where you normalize the relative frequencies in order to get the total area one. How do you construct this column? The column for the data should not be changed. Draw the bar chart!

Exercise 2

The function for the normal distribution is given by

\[f(x)=\frac{1}{\sigma \sqrt{2\pi}}e^{\left( - \dfrac{(x-\mu)^2}{2\sigma^2}\right)} \]

Use the mean 173.2492 and the standard deviation 14.0333 to plot the graph of the normal distribution.