For example we are to present the number of business calls made in a November (Table A) and the amount of donations made to a charity (Table B).
Table A: Table B:
In Table A it can be observed that there is a gap between the upper limit of the first group which is 5, the lower limit of the second group which is 6 and this is consistent amongst the other groups. This indicates that the data has been measured as whole numbers. Table B on the other hand, shows that the first and last class has defined upper and lower limits. These are known as open-ended classes. It is also useful such as cases like these to know the class boundaries. These are applied when there is a gap between classes, the boundaries must be fixed at exact halfway along the gap. Therefore, the class boundaries for class 1-5 are 0.5-5.5.
Before anything else there are certain rules that needs to be followed in order to create grouped frequency distributions. Firstly, all values must only be contained in one class meaning overlapping classes must not occur. Classes, which have 1-5, 4-10, 9-15 is incorrect because it is only applicable for whole numbers and does not accommodate values with decimal numbers. Secondly, the classes must be in numerical order or size order. It is also advisable to have an average of 8-10 classes in total. Less than 5 would not provide sufficient information and more than 15 would be too much to comprehend. It is preferable for frequency distributions to have equal class widths throughout. Although if it is not possible, then classes with smaller or larger widths can be used. Application of open-ended classes may only be used at the opposite ends of a distribution. If two frequency distributions having the same class structure with different frequencies are being compared, it is best to express the frequencies as percentages (commonly referred to as Relative Frequency Distributions).
There are different ways of forming classes in frequency distributions as long as the correct procedures are followed. First, calculate the range of values of the data given by subtracting the lowest value from the highest value. Second is to divide the range by 10 and adjust it upward to get the standard class width. Common class widths are usually 1, 2 or 5 or its multiples. The next step is to make the distribution classes. Make sure that the classes are put in size order. Now a table can be made. As an example, the procedure is applied to the previous Table A:
Lowest Value = 5, Highest Value = 30. So the Range is 30-5 = 25
Class Width is 25/10 = 2.5. If adjusted upward the class width is 5. Therefore Table A has the correct class range and class width.
With the knowledge of frequency distributions, a cumulative frequency can be made. A cumulative frequency represents the number of items that have values either above or below a particular level. It is the frequency written as a running total of frequencies. Adding a cumulative frequency column on Table B can show this:
By creating the cumulative frequency column, a cumulative frequency curve can be made. A cumulative frequency curve basically demonstrates how the values for the data given accumulate. The cumulative frequency is always on the Y-axis or the vertical axis while the intervals are on the X-axis or the horizontal axis of the graph. To form the cumulative curve, you basically plot the relevant points and connect these points with a straight line. This curve is commonly known as an ogive. Constructing such a graph also provides an easier method of presenting the data given. The graph below shows the cumulative frequency curve for Table B:
Besides graphically presenting the cumulative data, it also has another use. Through the graph, the median can easily be determined. The median is the middle value of the data. When collecting data, the median is found by arranging the data in size order, from smallest to biggest in order to find its midpoint. Calculating without the graph requires the use of a mathematical formula. Through the cumulative frequency curve, all that needs to be done is to look for the middle of the Y-axis.
Other relevant information that can be found on the cumulative frequency curve are the quartiles. There are two types of quartiles, the first or lower quartile and the third or upper quartile. The lower quartile is ¼ or 25% of the total frequency. The upper quartile is ¾ or 75% of the total frequency. In order to determine these quartiles, just apply the same technique used for the median and look ¼ and ¾ up the Y-axis. The graph below shows the median and the quartiles of Table B:
It can be observed that the median, just by looking at the cumulative curve, is exactly $425.00. With a more detailed graph the 1st and 3rd quartile can be determined. We know that the 1st quartile lands on $345.00 by looking the curve. It can also bee seen that the 3rd quartile lands close to $500.00. This kind of information can be very useful because it could say a lot about what it represents. For example, the foundation had 100 establishments donate funds. They wanted to find out what is the least amount of funds they can gather after the first 25% 100 establishments has donated money. This can be answered by the 1st quartile data. It can then be said that the least amount of funds that can be gathered after the first 25% has made a donation is $345.00. If the graph was read in reverse, the foundation can also see the most amount of funds they could gather after the first 25% of establishments has donated. This is through the use of the 3rd quartile. The most amount of funds they can gather after the first 25% of establishments has donated is close to $500.00.
After the given examples, it can be concluded that the frequency table and the cumulative frequency graph are very useful. The frequency table clearly presents how many times an event occurs and the number of items having values in a group. The cumulative frequency column shows the running total of the frequency. From the frequency table, a cumulative frequency curve can be made. This is also known as the ogive. The cumulative curve graphically presents the running total and from this curve we can determine the median, the first and third quartile and determining these are a lot easier graphically than manually or through a mathematical formula. Such kind of statistical information can be very useful in finding out more about the given data, as demonstrated in the Table B examples. Statistical information needs to be precise but it also needs to be understood easily. It is a lot easier to comprehend with the use of graphs and tables and part of seeking and calculating accurate information is accurate communication. Every statistical data has a purpose or use and its important to present its purpose well and so other people who needs the data can comprehend it. With the use of the frequency table and cumulative frequency curve, this can be achieved.
