Frequency (count): the number of times a particular value occurs.

A frequency distribution lists each distinct value with its frequency.

A bar plot is a graphical representation of a frequency distribution. Each bar’s height is based on the frequency of the corresponding category.

Relative frequency is the ratio of the frequency to the total number of observations.

\[ \text{relative frequency} = \frac{\text{frequency}}{\text{number of observations}} \]

This is also called the proportion.

The percentage can be obtained by multiplying the proportion by 100.

A relative frequency distribution lists each distinct value with its relative frequency.

A dot plot shows a number line with dots drawn above the line. Each dot represents a single observation.

• We would also like to be able to visualize larger, more complex data sets.
• This is hard to do using a dot plot!
• Instead, we can do this using bins, which group numeric data into equal-width consecutive intervals.

A random sample of weights (in lbs) from 12 cats:

\[\quad 6.2 \quad 11.6 \quad 7.2 \quad 17.1 \quad 15.1 \quad 8.4 \] \[\quad 7.7 \quad 13.9 \quad 21.0 \quad 5.5 \quad 9.1 \quad 7.3 \]

• Minimum is 5.5
• Maximum is 21

Lots of ways to break these into “bins”, but what about…

• 5 - 10
• 10 - 15
• 15 - 20
• 20 - 25

We’ve suggested bins

• 5 - 10
• 10 - 15
• 15 - 20
• 20 - 25

Each has an equal width of 5 (that’s good), but if we had a cat with a weight of exactly 15 lbs, would we use the second or third bin??

To make this clear, we need there to be no overlap. Instead, we could use:

Now, a cat with a weight of 15.0 lbs would be placed in the third bin (but not the second).

We will visualize this using a histogram, which is a lot like a bar plot but for numeric data:

This is a frequency histogram because each bar height reflects the frequency of that bin.

We can also create a relative frequency histogram which displays the relative frequency instead of the frequency:

Notice that those last two histograms look the same except for the numbers on the vertical axis!

• This gives us insight into the shape of the data distribution, literally how the values are distributed across the bins.

• The part of the distribution that “trails off” to one or both sides is called a tail of the distribution. - When a histogram trails off to one side, we say it is skewed.

  • right-skewed if it trails off to the right
  • left-skewed if it trails off to the left

• Data sets with roughly equal tails are symmetric.

We can also use a histogram to identify modes.

For numeric data, especially continuous variables, we think of modes as prominent peaks.

Finally, we can also “smooth out” these histograms and use a smooth curve to examine the shape of the distribution.