Ogive

Ogives with Ungrouped Data

Ungrouped data refer to raw data that have not been classified into categories. For example, the scores obtained by students on a final exam are considered in Table 1.

A frequency table is needed to construct an ogive from the raw data. The frequency table may include the cumulative frequencies, the relative cumulative frequencies, and/or the percentage cumulative frequencies for the analyzed data. In the case of the scores obtained by students on a final exam, the cumulative frequencies show the number of students who scored up to a particular number of points. The relative/percentage cumulative frequencies show the proportion/percentage of students who scored up to a particular number of points (Table 2).

The cumulative frequency distribution is obtained by adding the frequency corresponding to each score (column 2 in Table 2) to the sum of the frequencies of all smaller scores (column 3 in Table 2). For example, the cumulative frequency for the score of 61 is 0 + 2 = 2, because this is the lowest score obtained by students. The cumulative frequency for the next score (65) is 2 + 2 = 4. The next cumulative frequency (corresponding to a score of 75) is 4 + 1 = 5, and so on. This means that, for example, 5 students scored 75 points or less on the final exam, whereas 18 students scored 95 points or less.

The relative cumulative frequencies are obtained by dividing each cumulative frequency by the total number of observations (20 in this example). The percentage cumulative frequencies are obtained by multiplying the relative cumulative frequency by 100. In relative cumulative frequency terms, it can be concluded from Table 2 that, for example, [Page 960]a proportion of 0.20 of students scored 65 points or less on the final exam. Expressed in percentage, 20% of the students scored 65 points or less.

Figure 1 Ogive of the Scores Obtained by Students on a Final Exam, Using Ungrouped Data

The cumulative frequency of the highest score always equals the total number of observations (the sum of all frequencies, 20 in this example). The corresponding relative cumulative frequency is always 1.00; the corresponding percentage cumulative frequency is always 100%.

An ogive for the data in Table 2 is drawn using the scores obtained by students on the exam on the x-axis and either the cumulative, the relative, or the percentage cumulative frequencies on the y-axis (Figure 1).

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