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Types of Continuous Series

4.7 Types of Continuous Series — Master 6 types: Exclusive, Inclusive (with conversion steps), Open-End (3 cases), Cumulative (Less/More than), Unequal intervals, and Mid-Value conversion. CBSE Class 11 Statistics notes.

Notes

Types of Continuous Series

Class 11 Statistics — All 6 Types of Continuous Frequency Distributions

Types Overview

There are six types of continuous series. Click any type to jump directly to its detailed explanation below.

TypeDescription
Exclusive SeriesUpper limit excluded from its class
Inclusive SeriesBoth limits included in the class
Open-End DistributionLimit of first/last class not given
Cumulative FrequencyLess than and More than running totals
Unequal IntervalsClass-intervals not equal across classes
Mid-Value SeriesMid-values given instead of limits

Exclusive Series

Exclusive Series
In an exclusive series, the upper limit of one class becomes the lower limit of the next class. Such classification ensures continuity of data. For example, if lower limit is 10 and its upper limit is 20, then the class will be 10–20. It includes all values more than or equal to 10 but less than 20. The value 20 and above will go into the next class (20–30).

Example: Marks Distribution

ClassFrequency
10–206
20–305
30–409
40–5010

Boundary Diagram: Where does 20 belong?

10–20
20–30
30–40
40–50
Value 20 belongs to 20–30, NOT 10–20

Inclusive Series & Conversion

Inclusive Series
In an inclusive series, both lower and upper limits of a class-interval are included in the interval itself. Overlapping of intervals is avoided. For example, class 10–19 includes all items more than or equal to 10 and less than or equal to 19.
Class interval
6
5
9
10
10–20
20–30
30–40
40–50

Exclusive: Bars are touching because the upper limit of one class equals the lower limit of the next. There is no gap — data flows continuously from one class to the next. The bar width represents the full class interval (e.g., 10–20 has width 10).

Exclusive vs Inclusive — Comparison

Exclusive Method vs Inclusive Method
AspectExclusive MethodInclusive Method
Boundary RuleUpper limit counted in next classBoth limits counted in the same class
Upper/Lower LimitsUpper limit = lower limit of next classUpper limit differs from next lower limit (usually by 1)
Conversion Needed?No conversion needed prior to calculationMust convert to exclusive for simplicity in calculation

Open-End Distribution

Open-End Distribution
If the lower limit of the first class and the upper limit of the last class is not given, it is known as open-end distribution. Words like ‘below’ or ‘less than’ are used for the first class; words like ‘over’, ‘above’ or ‘more than’ are used for the last class.
Sometimes a dataset has extreme values. If we try to cover them with normal class-intervals, we may end up with too many classes, making the table lengthy. Open-end classes handle this compactly.
Mid-values cannot be accurately obtained, creating problems in computation of statistical measures. Also creates problems in graphic presentation.

Three Cases for Conversion

ClassWidth
Less than 10?
10–2020
20–3030
30–4040
Above 40?

Cumulative Frequency Series

A simple frequency distribution shows how many observations fall in each class. But sometimes we want the total number of observations up to or beyond a certain value. That is what cumulative frequency tells us.

2
5
10
12
17
4
0–10
10–20
20–30
30–40
40–50
50–60

Click a bound to see the running total:

Accumulate from left

2 + 5 + 10=17

17 students scored less than 30 marks

‘Less Than’ cumulative = add frequencies of all classes beforethe bound. ‘More Than’ cumulative = add frequencies of all classes after the bound. The total N = 50 (sum of all frequencies).

Reverse: Cumulative → Original Frequency Distribution

What if you are given a cumulative frequency distribution and need to find the original (simple) frequencies? The reverse process is: subtract consecutive cumulatives.

Chart A: Given Cumulative→Chart B: Original (derived)

Less Than Cumulative

2
7
17
29
46
50
Less than 10
Less than 20
Less than 30
Less than 40
Less than 50
Less than 60

Original Frequency

2
5
10
12
17
4
0–10
10–20
20–30
30–40
40–50
50–60

How the bars relate

Click “Next Step” to see the formula revealed one row at a time. Watch how each cumulative bar gets converted back into the original frequency bar.

From ‘Less Than’ CF:
First class f = first CF itself.
Every subsequent class: f = CFₙ − CFₙ₋₁
Verify: sum of all f = last CF = N.

From ‘More Than’ CF:
First class f = N − CF₂ (second CF).
Every subsequent class: f = CFₙ₋₁ − CFₙ
Verify: last class f = its own CF.

Quick check

Given “Less than 20” = 7 and “Less than 30” = 17, what is the original frequency for the class 20–30?

Answer: 17 − 7 = 10 (the number of observations that fall in the 20–30 range)

Unequal Class-Interval Series

Unequal Class-Interval Series
When the class-intervals of a frequency distribution are not equal, it is called unequal class-interval series. It happens when data is grouped in a way that some classes cover a larger range and some a smaller range, usually to simplify the presentation of data or to handle extreme values.
Class-IntervalFrequency
10–206
20–4015
40–7012
70–804
80–1103

Mid-Value Series

Mid-Value Series
Mid-value or Mid-point is the middle value of a class-interval. When such mid-values are given instead of class limits, it is called a mid-value series.

Mid-Value Converter

Difference between consecutive mid-values = 10

Half-difference = 10 / 2 = 5

Mid-ValueLower Limit (Mid − 5)Upper Limit (Mid + 5)
5010
151020
252030
353040
454050

Resulting class intervals: 0–10, 10–20, 20–30, 30–40, 40–50

Key Takeaways

Key Takeaways

  • Exclusive Series: upper limit excluded from its class (10–20, 20–30). Ensures continuity.
  • Inclusive Series: both limits included (10–19, 20–29). Convert to exclusive by adjusting boundaries by half the gap.
  • Open-End Distribution: first or last class limit is missing. Can sometimes be determined from the pattern of class widths.
  • Cumulative Frequency: Less Than keeps adding frequencies (bucket filling); More Than keeps subtracting (bucket draining).
  • To recover original frequencies from cumulative: subtract consecutive cumulatives (CFn − CFn−1).
  • Unequal Class-Interval Series: classes have different widths, used when data has extreme values.
  • Mid-Value Series: mid-values given instead of limits. Convert by subtracting/adding half the difference between consecutive mid-values.