Grouped data and bar graphs

Raw data is data exactly as collected — unsorted, unsummarised, hard to interpret directly. The marks of 30 students, listed in the order the answer sheets were checked, are raw data. The first job in data handling is to organise this pile into something a reader can scan in five seconds.

Class intervals do that job. You divide the full spread of values into equal-sized ranges, such as $0\text{–}10$, $10\text{–}20$, $20\text{–}30$, and slot every data value into exactly one range. Each interval has a lower limit and an upper limit; their difference is the class size:

where: the class size is the width of each bin, and at this level every bin in a table is kept the same width, so one number describes the whole table. By the exclusive convention used in ICSE Class 8, an interval like $20\text{–}30$ takes every value from $20$ up to but not including $30$; a value of exactly $30$ belongs to the next bin, $30\text{–}40$. Fixing this rule before you tally is what stops a boundary value from being counted twice — or missed altogether. For each interval, count how many raw values fall inside it (a tally-mark column keeps this honest); that count is the frequency of the interval. Listing frequencies against every class interval gives the grouped frequency distribution table — the organised version of the same raw data. A bar graph then draws that table. Every category or class interval becomes a rectangular bar of the same width; the height of a bar is proportional to its frequency, bars sit with equal gaps between them, and both axes carry a label and a scale starting at zero.

The goal of this derivation is to turn two raw facts about a data set — its lowest and highest values, and the class size you choose to work with — into the number of class intervals your table must contain, without missing the highest value or leaving an interval empty of purpose.

Step 1. Find the range of the raw data: $\text{Range} = \text{Maximum value} - \text{Minimum value}$. This is the total spread that every class interval together must cover, with no gaps and no overlap.

Step 2. Choose a class size $c$ — usually a round number like $5$, $10$ or $20$, chosen so the table has a manageable number of rows (roughly $5$ to $10$ intervals is the usual target for a school-level table).

Step 3. Divide the range by the class size to get the number of intervals needed: $n = \dfrac{\text{Range}}{c}$. If this division is not exact, round up to the next whole number, because a partial interval still needs a full bin to hold its values — you cannot cover the highest value with less than a whole interval.

Step 4. Starting from a convenient value at or just below the minimum, mark off $n$ intervals of width $c$ each. Check that the last interval's upper limit is strictly greater than the maximum data value — if it is not, add one more interval. This guarantees every single data value, including the extremes, has exactly one interval to belong to.

The marks (out of 100) obtained by 20 students in a test are:

Group this data into class intervals of size 10, starting from 30, draw the frequency table, and state how many intervals are needed.

Step 1. Find the range: minimum $= 39$, maximum $= 95$, so $\text{Range} = 95 - 39 = 56$. With class size $c = 10$, the derivation above gives $n = 56/10 = 5.6$, rounded up to $6$ intervals — but since we must also start cleanly at $30$ (below the minimum), the working table uses $7$ intervals from $30$ to $100$ to keep round numbers: $30\text{–}40, 40\text{–}50, 50\text{–}60, 60\text{–}70, 70\text{–}80, 80\text{–}90, 90\text{–}100$.

Step 2. Tally each value using the exclusive convention (a value equal to an interval's upper limit goes to the next class): $39$ is in $30\text{–}40$; $45, 48, 42$ are in $40\text{–}50$; $52, 58, 55, 50$ are in $50\text{–}60$; $68, 62, 66, 64, 60$ are in $60\text{–}70$; $74, 71, 77$ are in $70\text{–}80$; $81, 83$ are in $80\text{–}90$; $90, 95$ are in $90\text{–}100$.

Step 3. Verify the total: $1+3+4+5+3+2+2 = 20$, matching the 20 students exactly. This check is the single most reliable way to catch a mistallied value before the table is used to draw anything.

Step 4. Draw the bar graph: class intervals along the horizontal axis, frequency on the vertical axis (scale: 1 unit = 1 student), one bar per interval, all bars the same width with equal gaps. The tallest bar, height 5, sits over 60–70 — the most common scoring range in this test.

• Boundary confusion — forgetting which interval a value like exactly 50 belongs to; fix one convention (lower limit stays, upper limit moves to the next class) and apply it every time.
• Unequal bar widths in the bar graph, which visually exaggerates or hides differences — every bar must be the same width, with frequency shown only through height.
• Skipping the frequency-total check — always add the frequency column and confirm it equals the total number of observations before trusting the table.
• Starting the frequency axis above zero or leaving gaps uneven, both of which make bar heights misleading to read at a glance.

Ask the tutor: if you are stuck choosing class intervals for your own raw data set, or want your bar graph checked for equal widths and correct labels, ask the AI tutor to walk through it with you step by step.