Grouping data: classes, frequencies and ogives

Detail

Why we group data

• Raw data: figures written down as they were collected, before any sorting. NCERT's example is the geography scores of 60 students.
• Raw data is bulky, and you cannot see any pattern in a long list of numbers.
• Grouping of data: sorting raw data into classes (groups that each cover a range of values) so that there is less of it to handle. NCERT calls this the process of classification.
• Range: the highest value minus the lowest value. NCERT's 60 scores run from 02 to 96.
• Class interval: the upper limit of a class minus its lower limit. In the class 20–30, the class interval is 30 − 20 = 10.
• The range decides how many classes you need and how wide each one is:
• range 02–96 → about 100 units to cover

• 100 ÷ 10 → ten classes of 10 (0–10, 10–20 … 90–100) (Tables 1.4, 1.5)

• Class limits: the lowest and highest values a class can hold, called the lower limit and upper limit.

• Class midpoint (mid-value) = (lower limit + upper limit) ÷ 2. For 20–30 it is 25. Graphs and averages use it as the value for the whole class.

Official data comes grouped too

• The Census groups age data the same way. Census 2011 Table C-14 is called "Population in five year age group by residence and sex" [2].
• Its classes are 0–4, 5–9, 10–14 … 75–79, then an open class 80+, plus "age not stated" [2].
• Each class covers 5 years of age. The classes are written the inclusive way (0–4, then 5–9), which is explained below.

• The table is given for India, states, districts and cities, with separate figures for total, rural and urban people [2].

• An open-end class such as "80+" has no upper limit. It is used when a few values are very high and spread far apart.

Tally marks: counting each item

• Tally marks: one short stroke for each item, drawn against the class the item falls into.
• "Four and cross" method: draw four upright strokes. The fifth stroke goes across those four (𝍸). This makes a bundle of five.
• Why it helps:
• you count in fives, so counting is quick
• each item is marked once, so it is not missed and not counted twice
• you can check that all the bundles add up to N (60)

Frequency distribution

• Frequency (f): the number of items that fall in one class.
• Frequency distribution: a table that lists every class with its frequency.
• Simple frequency: the count for each class alone.
• The frequencies of all the classes add up to the total number of items: Σf = N = 60. (Σ, "sigma", means "sum of".)
• A frequency distribution shows where values crowd together and where they thin out. A raw list cannot show this.

Cumulative frequency

• Cumulative frequency (Cf): a running total. You add each class's frequency to the totals of all the classes before it.
• It answers "how many items are below this value?"
• NCERT's example:
• 27 students scored less than 50

• 45 students scored less than 70

• The last Cf always equals N (60). This is a quick way to catch an adding mistake.

• From a less-than total you can get a "more than" total: more than L = N − (less than L).
• Less than 50 = 27 → more than 50 = 60 − 27 = 33
• Less than 70 = 45 → more than 70 = 60 − 45 = 15

Exclusive and inclusive methods

Feature	Exclusive method	Inclusive method
How classes are written	20–30, 30–40, 40–50	20–29, 30–39, 40–49
Limits	The upper limit of one class is the lower limit of the next	Both limits belong to the class
Boundary value (30)	Goes to the class where it is the lower limit, so 30 is in 30–40	Belongs to only one class, so the question does not arise
How it is read	"20 and under 30"	"20 to 29, both included"
Next class starts	At the same value (30)	One unit higher (30, after 29)
Values in each class	10	10 (20, 21 … 29)
Best for	Continuous data (data that can take any value, including decimals), e.g. rainfall in mm, temperature in °C	Discrete data (data that takes only whole numbers), e.g. marks, number of villages, age in completed years
Example	Score classes 0–10 … 90–100	Census 2011 age groups 0–4, 5–9 … [2]

• In the exclusive method, the upper limit is left out of the class. So 29.9 falls in 20–30, but 30 falls in 30–40.
• In the inclusive method, there is a gap of one unit between classes (29 and 30). Both methods still give each class ten units.
• Converting inclusive classes to exclusive ones (beyond NCERT): take half the gap (0.5) away from each lower limit and add it to each upper limit. So 20–29 becomes 19.5–29.5. This is needed before you draw graphs, because the classes must touch with no gap.

Frequency polygon

• Frequency polygon: a line graph of a frequency distribution.
• How to draw one:
• put the classes on the x-axis and the frequency on the y-axis
• plot each class's frequency, usually at the class midpoint
• join the points with straight lines

• close the shape by meeting the x-axis at the midpoints of empty classes at each end

• Use: draw two distributions on the same axes to compare them, for example the scores of two subjects or of two years (Fig. 1.5). Two overlapping histograms (bar graphs of a frequency distribution) are hard to read. Two lines are easy to read.

• The other graph types are covered in the data-representation and thematic-maps notes.

Ogive (cumulative frequency curve)

• Ogive: a curve that plots cumulative frequencies against class limits (Fig. 1.6–1.8).

Feature	Less-than ogive	More-than ogive
Plotted against	Upper limits (less than 10, less than 20 …)	Lower limits (more than 0, more than 10 …)
Starts at	Near 0	N (60)
Shape	Rises from left to right	Falls from left to right
Ends at	N (60)	Close to 0
Answers	"How many are below x?"	"How many are above x?"

• Median: the middle value, with half the items below it and half above.
• Where the two ogives cross, the x-value is the median, and the y-value is N/2 (here 30). This goes beyond what NCERT says.
• Checking this with NCERT's data:
• N/2 = 30. Thirty students are below 50 + something.
• 27 students are below 50. The 50–60 class has 33 − 23 = 10 students.
• Median = L + [(N/2 − Cf) ÷ f] × i = 50 + [(30 − 27) ÷ 10] × 10 = 53 (L = lower limit of the median class, Cf = cumulative frequency before it, f = its frequency, i = class interval).

Error in NCERT Table 1.9

• The "more than" figures in Table 1.9 of NCERT are wrong from "more than 30" onwards.

Class (more than)	NCERT's figure	Correct figure
More than 30	44	46
More than 40	38	39
More than 50	28	33
More than 60	20	23
More than 70	14	15

• How to check: use more than L = N − (less than L).
• Less than 50 = 27 → more than 50 must be 60 − 27 = 33, not 28.

• Less than 70 = 45 → more than 70 must be 60 − 45 = 15, not 14.

• Because the more-than ogive in the book is drawn from wrong numbers, it falls too low. So it crosses the less-than ogive at the wrong place, and the median read from the crossing point is also wrong.

Prelims Hooks

• Class interval = upper limit − lower limit. For 20–30 it is 10.
• In the exclusive method, a boundary value such as 30 goes to 30–40, the class where it is the lower limit, not to 20–30.
• In the inclusive method (20–29, 30–39), both limits are part of the class, and the next class starts one unit higher.
• "Four and cross": the fifth tally stroke crosses the four before it.
• Σf = N, and the last cumulative frequency always equals N.
• Less-than ogive → upper limits → rises. More-than ogive → lower limits → falls.
• The two ogives cross at the median, at a height of N/2. They do not cross at the mean or the mode.
• A frequency polygon is plotted at class midpoints and is the best simple graph for comparing two distributions.
• Census 2011 Table C-14 groups people into five-year age groups (0–4 … 75–79, then 80+) [2].

Mains Points

• Grouping turns raw data into something you can use for planning. Census 2011's five-year age groups by rural and urban areas, down to district level [2], show where there are many children (who need schools) and many old people (who need health care). District plans use these numbers.
• Choosing classes involves a trade-off. Too few classes hide differences between regions. Too many classes leave the data almost as bulky as the raw list. So the class interval should be chosen from the range of the data, as with ten classes of 10 for scores from 02 to 96.
• Ogives make comparison between regions easy. You can read the median, or the share of units below a given value, straight from the curve. Examples are the share of districts below a literacy level or below a rainfall level.
• Data quality matters. One wrong cumulative figure, like those in NCERT Table 1.9, moves the median. Simple checks, such as last Cf = N and more-than = N − less-than, should be part of every step of data processing.