Collection and Types of Data

Before we can compute means or draw graphs, we need data , numbers (or observations) collected systematically. This lesson clarifies what kinds of data exist, how they are collected, and the standard vocabulary statisticians use.

Definitions

Data is a collection of facts, observations, or measurements. Each individual observation is a datum.

Primary data is collected directly by the investigator , by survey, experiment, or measurement. Example: a student measures the heights of all classmates.

Secondary data is obtained from other sources , books, government publications, websites, previous surveys. Example: looking up census data published by the government.

Ungrouped (raw) data is a list of individual values, e.g. "test scores: 56, 78, 90, 45, 67, 88, ...".

Grouped data is organised into intervals (or classes), e.g. "scores in 0-20: 5 students; 20-40: 12 students; ...".

Concepts

Range. The difference between the largest and smallest values in a data set: $\text{Range} = \text{max} - \text{min}$.

Class interval. When organising data into groups, each group has a range like "10-20" or "20-30". The width of each class is the class size (or class width).

Frequency. The number of times a particular value (or class) appears in the data.

Class limits. The lower limit and upper limit of a class. For class "20-30", lower limit is 20, upper limit is 30.

Class mark (midpoint). The middle value of a class: $\dfrac{\text{lower limit} + \text{upper limit}}{2}$. For class 20-30, class mark is 25.

Cumulative frequency. The running total of frequencies. Useful when finding the median for grouped data (in higher classes).

Choosing class size

When grouping data, choose the class size so that:

• There are 5 to 15 classes (too few hides patterns; too many makes the picture noisy).
• The class size is a "nice" number like 5, 10, 20.
• The first class starts at or just below the minimum value.

For example, if data ranges from 23 to 89, you might choose classes 20-30, 30-40, ..., 80-90. Class size 10, eight classes total.

Inclusive vs. exclusive class intervals

Exclusive form: intervals like 0-10, 10-20, 20-30. A value of exactly 10 goes in the 10-20 interval (not 0-10). This is the standard form for continuous data.

Inclusive form: intervals like 0-9, 10-19, 20-29. A value of 10 goes in 10-19. Useful for discrete data (like counts).

Both forms have the same total frequencies; they just label the boundaries differently.

Worked examples

Example 1. Range of the data $4, 8, 12, 15, 20, 25$.

Range $= 25 - 4 = 21$.

Example 2. A class has 30 students with test scores. The teacher organises them into classes 0-20, 20-40, 40-60, 60-80, 80-100. State the class size and the class marks.

Class size: 20. Class marks: 10, 30, 50, 70, 90.

Example 3. The marks of 20 students are: 12, 15, 17, 18, 20, 22, 25, 25, 28, 30, 32, 33, 35, 36, 38, 40, 42, 45, 48, 50. Group into classes of width 10 starting at 10.

(Note: value 20 is in 20-30 not 10-20 by the exclusive convention. Similarly 30, 40, 50 each move up to the next class.)

Example 4. From a list of daily temperatures over a week, identify whether the data is primary or secondary.

Primary , if you measured each temperature yourself. Secondary , if you obtained them from a weather website.

Example 5. A frequency distribution has classes 20-25, 25-30, 30-35, 35-40 with frequencies 5, 7, 4, 2. Find the cumulative frequencies and the total.

Cumulative: 5, 12, 16, 18. Total: 18.

Try it yourself

1. State the difference between primary and secondary data.
2. Find the range of 25, 32, 47, 56, 18, 91.
3. A frequency distribution has classes 0-10, 10-20, 20-30 with frequencies 8, 12, 5. Find the class size and the class marks.
4. The marks of 15 students are: 10, 14, 18, 22, 23, 27, 31, 32, 36, 39, 40, 42, 45, 48, 50. Group into classes of size 10 starting at 10.
5. Distinguish between inclusive and exclusive forms.
6. Find the cumulative frequency for the distribution of question 3.
7. A survey of pet ownership in 30 households shows: 1, 0, 2, 1, 3, 0, 1, 2, 4, 1, 0, 1, 1, 2, 3, 1, 0, 1, 2, 1, 0, 1, 2, 3, 1, 0, 1, 2, 1, 4. Make a frequency table.
8. State the class mark of the interval 50-60.
9. Why must the class width be uniform?
10. Identify the primary data source you would use to study the heights of students in your school.

Pitfalls / Insight

• Exclusive vs. inclusive. Be consistent with one convention throughout a problem.
• Class marks are midpoints, not endpoints.
• Range is just $\max - \min$. No frequency involved.

Insight. Good data organisation is the foundation of good statistics. Once your frequency table is clean, every graph and summary calculation falls out naturally. Spend time on the table.