Frequency Distributions A frequency distribution of a dataset is a
display of its distinct categories along with the number of observations in each
category.
The goal of creating frequency distributions is to present raw data
in more meaningful ways.
It involves organizing and displaying data so we can derive useful information at a quick
glance.
Consider this data of blood types for 20 patients at a hospital.
We can organize them by categories in a frequency table as seen here.
Because the data are non-numeric (not numbers), we refer to them as qualitative or categorical
data.
The frequency is the number of times each category occurs.
This table is an example of a frequency distribution because it shows how the frequencies are distributed
across the categories.
So, to have a frequency distribution, we need the categories, or groups, or classes, or
intervals and the frequency of occurrence of these intervals.
And we can extend this to what we call relative frequencies by dividing each frequency by
the total frequency.
Relative frequencies can be expressed as decimals or precents.
And they should add up to 1 or 100%.
Now, because the variable is categorical, commonly used graphical representation are
bar charts and pie charts.
While bar charts use bars to display either frequencies or proportions in each category,
pie charts mainly use slices of pies to display proportion that each category represents relative
to the whole.
And when we organize the bars in a bar chart in descending order of magnitude,
we refer to it as a Pareto chart.
Now, consider this dataset consisting of the ages of 25 patients at a hospital.
We can attempt constructing the frequency distribution by listing all the values from
12 all the way to 56 but that will even be less meaningful because the range of the data
is wide.
This approach could be useful when there are only few different data points with a lot
of repetition.
A more suitable approach is to organize this into a grouped frequency table
by creating class intervals and listing the number of observations in
each interval.
For example, we have 2 observations from 10 to 19,
and we have 5 of them from 20 to 29.
If the data values are all discrete as shown here, this table format is acceptable, but
if the values are continuous or take on decimals, these intervals will no longer work.
For example, values like 19.5 or 29.35 will not be captured by any interval listed here.
As a result, we often group the data into class intervals with open ended upper limit
such as “10 to under 20”, “20 to under 30”, and so on.
This first interval will start at 10 and go on to 19.9999… and the second interval will
start at 20 and go on to 29.9999…
Although this second presentation is more useful for continuous data, it still works
for discrete data as well.
We will refer to the starting values as the lower limits and ending ones the upper limits.
Note that according to our definition here, the upper limits are not included in their
respective intervals.
Now, we have what is called the class width or the width of the class interval.
This is defined as the difference between two consecutive lower limits
or difference between two consecutive upper limits.
Another way to look at it is this: If the values are discrete, from 10 to 19,
we have 10 discrete points.
But if they are continuous, we have 10 units of equal distances between 10 and 20.
Therefore, either way, the class width is 10 for these two presentations.
Now, we sometimes need to perform calculations on class intervals that require choosing a
representative from each interval.
The best choice is usually the class midpoint which can be found by taking the median or
average of the lower and upper limit for each class interval.
For the first class interval, the midpoint is found by taking 10 + 20 and dividing by
2, which gives 15.
We obtain the other midpoints respectively.
Here again are the frequencies.
To graphically present frequency distribution with class intervals, we often use a histogram.
A histogram is essentially a bar graph where the horizontal axis represents continuous
intervals.
The height of the bars still represents frequency.
There is usually no space between the bars unless the frequency on an interval is zero.
Unlike bar charts, if we change the order of the bars in a histogram, it will no longer
be meaningful.
The class with the highest frequency is called the modal class while the class containing
the middle value in the sorted data is the median class.
That is, the median class is the “40 to under 50 “class because it contains the
middle observation which is the 13th observation.
Note that the median class is not necessarily the middle class, as in 30 to 40 in this case,
but it is the class that contains the middle observation which is the 13th observation
and that is contained in the 40 to under 50 class.
We can obtain another graph by connecting the midpoint of the bars with straight lines
and tying them to the horizontal axis.
This graph is called a frequency polygon.
Another way to extract useful information from data is by using cumulative frequencies.
A cumulative frequency represents the sum of all frequencies up to the class interval
in question.
Note that our upper limits here are not included in the respective class intervals.
Thus, these cumulative frequencies tell us how many values are below the upper limit
of a class.
To calculate cumulative frequencies, we start with the frequency of the first class, then
add the frequency of the second class to obtain the cumulative frequency for the second class.
Then add the frequency of the third class to that to obtain the cumulative frequency
of the third class.
And so on.
The cumulative frequency of the last class will always be equal to total frequency.
We can also calculate cumulative relative frequency by dividing the cumulative frequencies
by total frequency.
And here are the results as percents.
We see here that 28% of the data are less than 30, and 64% of the data are less than
50.
Graphically, the cumulative frequency distribution is presented in a curve called ogive.
The graph starts at a frequency of zero for the lower limit of the first class and the
cumulative frequencies are plotted against the upper limit of each class.
We can now use the graph to estimate the proportion of the observations below a value.
For example, we can estimate that the proportion of the data that are less than 35 is about
40%.
Finally, we often classify frequency distributions based on the shape of their histogram.
If a lot more observations cluster around higher values compared with lower values,
we say the distribution is negatively skewed.
That is, the tail of the distribution extends to the left.
An example will be scores on an easy exam.
The frequencies will be higher for larger scores than for smaller scores.
On the other hand, if the frequencies are substantially higher for lower scores than
for larger ones so that the tail extends to the right, we have a positively skewed distribution.
An example of that will be a difficult exam.
Income is also generally positively skewed.
The two distributions at the bottom here are called symmetric.
That is, if you cut them in half, the left side will be identical to the right side.
The one on the bottom left is referred to as a bell-shaped distribution while on the
bottom right is called a uniform distribution because the bars of the histogram all have
the same height, or the frequencies are all the same.
And that’s a very quick overview of frequency distributions.
Thanks for watching.