Lesson 5: Tabulation, Frequencies and Graphs (II)January 17, 2012
2. Tabulation. Frequency distribution table
In any statistical investigation, the collection of the numerical data is the first and the most important matter to be attended
2.1.Counting data: tabulation
This is the process of condensation of the data for convenience, in statistical processing, presentation and interpretation of the information.
A tally mark “|” is put next to the variable each time a value for that variable appears,
Frequency Distribution is a tallying of the number of times (frequency) each score value (or interval of score values) is represented within a group of scores.
A frequency distribution is ungrouped if the frequency of each score value or piece of data is given
2.2. Absolute frequency and relative frequency
Absolute frequency or frequency of a statistical datum, xi, is the number of times this datum appears. It is represented by fi.
fi. is the absolute frequency of xi
PROPERTY: The sum of the absolute frequencies of a set of statistical data is the total number of data, N.
f1.+ f2.+ ….+ fn.= N
The relative frequency of a statistical datum is the frequency of the datum divided by the total number of data. Many times it is expressed as a percentage. It is represented by hi
hi = fi./Nis the relative frequency of xi,
PROPERTY: The sum of the relative frequencies of a set of statistical data is equal to one.
h1.+ h2.+ ….+ hn.= 1
2.3 Cumulative Frequency
Cumulative absolute frequency of a datum xi, is the sum of all the absolute frequencies of all values less than or equal to . It is represented by Fi .
Fi = f1.+ f2.+ ….+ fi.
Cumulative relative frequency of a datum xi, is the sum of all the relative frequencies of all values less than or equal to xi,. It is represented by Hi
PROPERTY: The cumulative relative frequency of a datum xi,, Hi., is equivalent to the cumulative absolute frequency of the datum xi, divided by the total number of datum.
Hi.= h1.+ h2.+ ….+ hi = f1./N+ f2./N+ ….+ fi /N = Fi /N
To work out the cumulative frequencies the data must be ordered, this is the statistical variable must be quantitative.
On the directions below you can practice with absolute frquencies:
3. Statistical graphs
We can also organize data using graphs. Graphs are a visual method to summarize data and allow the observer to see the relevant features of the statistical study.
4.1 Bar charts
Bar charts or bar graphs consist of a vertical axis and a series of labelled or vertical bars that show the different values of the variable for each bar. The numbers along the vertical axis of the bar graph are called the scale and show the frequency of each value.
They are used when the variable has several different values.
They are used for qualitative variables and discrete quantitative variables (but ungrouped).
So we follow these steps:
- We write the values of the variable on the X axis
- We write the frequency with the appropriate scale on the Y axis.
- The frequency corresponding to a value is represented by a bar. The heights of the bars are proportional to the frequency.
When the variable is quantitative, the tops of the bars can be joined by segments obtaining a polygonal line named frequency polygon.
You can practice with bar charts on the pages below
The video below tries to teach you how to avoid misleadings in bar charts (or bar graphs)
3.2. Pie charts
A pie chart is a circle graph divided into pieces, or circular sectors, each displaying the size of some related piece of information. Pie charts are used to display the sizes of parts that make up some whole.
They can be used for any type of variable (qualitative or quantitative).
The measurement in grades of each circular sector is directly proportional to the frequency of the value of the variable that it represents.
If we set up a rule of three direct, we get the degrees corresponding to each xi:
Number of data Degrees
N ———————————– 360º
fi ——————————————— aº
aº = (fi / N) · 360º
This way the amplitude of the circular sector in degrees can be obtained by applying this formula:
Degree of the central angle of the sector = (fi / N) · 360º = hi
It is advisable to use the rule of three instead of the formula. You will need a protractor to draw pie charts.
On these links you can practice pie charts
The video below shows how to create a pie chart (or pie graph or circle graph!)