Q.Width of rectangles in a histogram should essentially be equal (True/False).
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Histogram Bin Width – A First Look
Imagine you're a Class 12 Economics student collecting data on the monthly pocket money of 50 students in your school. You have numbers like ₹200, ₹350, ₹500, ₹1200, and so on. If you just list them, you see nothing. But if you group them — say, ₹0–₹200, ₹200–₹400, ₹400–₹600 — you start to see a pattern: most students get between ₹200 and ₹600, a few get more.
That grouping interval — ₹200 in this example — is the bin width. In a histogram, the horizontal axis is divided into equal-sized bins (intervals), and the height of each bar shows how many observations fall into that bin. The bin width is simply the length of each interval.
The Precise Meaning
For a continuous variable (like income, price, age, or exam score), a histogram is a bar chart where:
- The width of each bar equals the bin width (say, h).
- The height of each bar is the frequency (count) of data points in that bin.
- The area of each bar is proportional to the relative frequency.
If you choose a bin width of ₹200, your bins might be:
₹0–₹200, ₹200–₹400, ₹400–₹600, ₹600–₹800, ₹800–₹1000, ₹1000–₹1200.
In Economics, we often use histograms to show the distribution of income, expenditure, prices, or marks. The shape of the histogram tells you whether most people earn a little (right-skewed) or a lot (left-skewed), or whether the data is symmetric.
Why Bin Width Matters
The bin width is not a fixed number — you choose it. And that choice changes the story the histogram tells.
- Too wide (say ₹500): you might get only 2–3 bars. You lose detail — you can't see whether most students get ₹200–₹400 or ₹400–₹600.
- Too narrow (say ₹10): you get many bars, each with very few students. The histogram looks jagged and noisy — you can't see the overall pattern.
- Just right (say ₹200): you see the shape clearly — a peak around ₹300–₹500, a tail towards higher values.
A common rule of thumb for choosing bin width is Sturges' rule:
k=1+log2n
where k is the number of bins and n is the number of data points. Then bin width = (max value – min value) / k. For 50 students, k≈1+5.64=6.64, so about 7 bins. If the range is ₹1000, bin width ≈ ₹143.
The Formula (Where It Exists)
In Economics, the histogram itself is a graphical tool — it doesn't have a single formula like the multiplier. But the bin width is defined as:
Bin width=Number of binsRange of data
where:
- Range = Maximum value – Minimum value
- Number of bins = chosen by the researcher (often using Sturges' rule or trial and error) …
In a histogram, the width of each rectangle is set by its own class interval, while it is the area of the rectangle — not the height alone — that must stay proportional to the frequency. …
False. In a histogram the width of a rectangle equals its class interval, so with unequal class intervals the widths are unequal. It is the area, not the height, that must stay proportional to frequency.
Concept first: area, not height, carries the frequency
In a histogram, each class is drawn as a rectangle whose width = the class interval and whose area = the class frequency. When all class intervals are equal, the widths are equal and heights are simply the frequencies. But when the class intervals are unequal, the widths differ.
Adjusting for unequal intervals
To keep areas comparable when widths differ, we compute an adjusted frequency (frequency density): …
- JKBOSE Class 11 (Commerce) 2020Set ANNUAL6 marksQ.Construct histogram and frequency polygon from the following data :
Marks Frequency 0—10 5 10—20 10 20—30 15 30—40 18 40—50 8 (OR)Explain the meaning of Sampling Survey. What are its advantages ?›Reveal solutionSolution
Since every class is continuous and of equal width (10), the histogram is drawn with adjoining bars whose heights equal the actual frequencies, and the frequency polygon is obtained by joining the midpoints of the tops of these bars (plus two zero-frequency points at each end) with straight lines.
Step 1 — Since the classes are already continuous with equal width (10), no adjustment to frequencies is needed; bar height = class frequency directly.
Marks (class, exclusive) Frequency Class midpoint 0–10 5 5 10–20 10 15 20–30 15 25 30–40 18 35 40–50 8 45 Step 2 — Constructing the histogram. Mark 'Marks' on the X-axis (class boundaries 0, 10, 20, 30, 40, 50) and 'Frequency' on the Y-axis. For each class, draw a rectangle whose base is the class width (0–10, 10–20, …, 40–50) and whose height equals its frequency (5, 10, 15, 18, 8). Since the classes are continuous and exclusive, the rectangles are drawn touching each other with no gaps, giving five adjoining bars of heights 5, 10, 15, 18 and 8.
Step 3 — Constructing the frequency polygon. Take the midpoint of each class (5, 15, 25, 35, 45) and plot a point at a height equal to that class's frequency: (5, 5), (15, 10), (25, 15), (35, 18), (45, 8). To close the figure on the X-axis, add one imaginary class of zero frequency immediately before the first class (i.e., −10 to 0, midpoint −5, frequency 0) and one immediately after the last class (50 to 60, midpoint 55, frequency 0). Join all these points — (−5, 0), (5, 5), (15, 10), (25, 15), (35, 18), (45, 8), (55, 0) — in order, using straight lines. (A frequency polygon can equally be drawn directly on top of the histogram, by joining the midpoints of the top of each bar.)
Shape read from the data: the polygon rises steadily from the lowest class up to a peak at the 30–40 class (frequency 18), then falls for the 40–50 class — showing that marks are concentrated in the 30–40 range, with the distribution being mildly skewed rather than symmetric.
…
- JKBOSE Class 11 (Commerce) 2019Set ANNUAL6 marksQ.Draw a histogram, a frequency polygon and frequency curve of the following data :
Marks 0–10 10–20 20–30 30–40 40–50 50–60 No. of Students 5 12 15 22 14 4 (OR)What do you mean by classification of Data ? Explain its various objectives.›Reveal solutionSolution
This question has an OR — Part (a) asks to draw a histogram, frequency polygon and frequency curve for the given grouped data; Part (b) (OR) asks what classification of data means and its objectives. Both are answered in full below.
Part (a): Histogram, Frequency Polygon and Frequency Curve
Data (continuous, equal class width = 10):
Marks 0–10 10–20 20–30 30–40 40–50 50–60 No. of Students 5 12 15 22 14 4 Step 1 — Histogram. Since the class intervals are continuous and of equal width, a histogram is drawn by marking class limits (Marks) on the X-axis and frequency (No. of students) on the Y-axis, then erecting adjacent rectangles over each class interval with height equal to its frequency (no gaps between bars, since the classes are continuous): bars of heights 5, 12, 15, 22, 14, 4 over 0–10, 10–20, 20–30, 30–40, 40–50, 50–60 respectively.
Step 2 — Frequency Polygon. Mark the mid-points of the top of each histogram bar — these are the class mid-points (5, 15, 25, 35, 45, 55) plotted against their frequencies (5, 12, 15, 22, 14, 4). Join these mid-points with straight lines. To close the polygon, extend it to the mid-points of two imagined classes with zero frequency at each end (i.e., the class before 0–10, which is −10–0 with mid-point −5, and the class after 50–60, which is 60–70 with mid-point 65), each plotted at frequency 0, and join these to the first and last points. This can be drawn either on the same axes as the histogram (joining bar mid-tops) or independently by plotting (mid-point, frequency) pairs directly.
Frequency polygon points (mid-point, frequency): (−5, 0), (5, 5), (15, 12), (25, 15), (35, 22), (45, 14), (55, 4), (65, 0)
Step 3 — Frequency Curve. A frequency curve is obtained by smoothing the frequency polygon into a free-hand continuous curve, instead of joining the points with straight lines — the curve passes as closely as possible through (or near) all the plotted points, showing the general shape/trend of the distribution rather than its exact day-to-day fluctuation.
All three diagrams use the same X-axis (Marks, class intervals/mid-points) and Y-axis (Number of students), only the method of plotting (bars / straight-line polygon / smoothed curve) differs.
OR — Classification of Data and its Objectives
Meaning: Classification of data is the process of arranging raw, unorganised data into different groups or classes according to similarities, resemblances, or common characteristics shared by the items, so that the data becomes easier to understand and analyse.
Objectives of classification of data:
- To condense the data — reduces a large mass of raw, scattered figures into a few meaningful, manageable groups/classes.
- To make data simple and comprehensible — organised data is easier to read, understand and compare than raw, unsorted data.
- To bring out similarities and dissimilarities — grouping similar items together and placing dissimilar items in different classes helps highlight patterns in the data. …
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