In making a frequency distribution, you set the real limits for the lowest two intervals at 29.5–39.5 and 39.5–49.5. A friend objects, saying, “The intervals overlap, because both contain 39.5.”
(a) Reply to your friend’s objection.
(b) Give the apparent limits and the interval width for both intervals.
The Correct Answer and Explanation is:
Correct Answer:
(a) Reply to your friend’s objection:
The intervals do not overlap, even though both appear to include 39.5. This is because real limits are used to define the boundaries between intervals without overlap by convention. In practice, the upper real limit of the first interval (39.5) is the same as the lower real limit of the second interval, but a score of exactly 39.5 is not counted in both; it belongs only to the second interval (39.5–49.5).
(b) Apparent limits and interval width:
- First interval:
- Apparent limits: 30–39
- Interval width: 10
- Second interval:
- Apparent limits: 40–49
- Interval width: 10
Explanation):
In statistics, particularly when creating frequency distributions for grouped data, it’s important to distinguish between apparent limits and real limits of a class interval.
Apparent limits are the whole number boundaries shown in the table (e.g., 30–39), while real limits adjust these boundaries to account for measurement precision (e.g., 29.5–39.5), allowing data to be grouped properly without overlap or gaps.
Your friend’s concern about overlapping intervals arises from the fact that both intervals seem to include 39.5. This is a common misunderstanding. However, when defining class intervals using real limits, we follow the convention that each value belongs to one and only one interval. By this rule, a score of 39.5 would be assigned to the second interval, not both. This system avoids double-counting and ensures that data are properly grouped.
This distinction becomes critical when the data are continuous, such as heights, weights, or test scores, where precision matters and values can fall exactly on boundaries like 39.5. Real limits are used to bridge the gaps between apparent limits, ensuring that each score has a precise place in the distribution.
To determine the interval width, you subtract the lower real limit from the upper real limit of a class (e.g., 39.5 – 29.5 = 10). Both intervals have the same width: 10. Apparent limits for 29.5–39.5 are 30–39, and for 39.5–49.5 they are 40–49.
In summary, there is no overlap, and using real limits properly helps in building accurate, non-redundant frequency distributions.
