Statistics

1. Primary vs Secondary Data

Data is any collection of facts, numbers, measurements or other information gathered for a purpose.

• Primary data: Information collected first-hand by the investigator themselves — through surveys, direct measurements, experiments. The collector knows the context, source and accuracy exactly. Example: a teacher recording attendance each day.
• Secondary data: Information taken from existing sources not originally created by the investigator. Examples: census reports, newspaper statistics, government publications, internet databases, published research.

Raw data is data that has just been collected, unsorted, and looks like a jumble of numbers. The first step in statistics is always to organise this raw data.

Arrayed data: When raw data is arranged in ascending or descending order it becomes an array. This is a necessary first step before constructing frequency tables or finding the median.

Range: The difference between the largest and smallest values in the data. It gives a rough idea of how spread out the data is.

2. Frequency Distribution — Tally Marks, Class Intervals, Class Width, Class Mark

A frequency distribution table lists each value (or range of values) with the number of times it occurs. This compresses raw data into a clear summary.

Tally marks

While building the table, we go through the data one value at a time. Instead of writing numerals, we draw tally marks — four vertical strokes, and the fifth diagonally crosses the bundle (like a "gate"). Counting bundles of five is fast and error-free. Each complete bundle represents 5.

Ungrouped frequency distribution

Lists every distinct individual value with its frequency. Best when the range of values is small (e.g., blood groups A, B, O, AB).

Grouped frequency distribution

Groups data into class intervals. Used when the range is large (e.g., marks 0–100). Each class interval has a lower class limit and upper class limit.

Exclusive (continuous) form vs inclusive form

Exclusive form: 10–20, 20–30, ... means a value of exactly 20 goes into 20–30 (upper limit excluded from the lower class). Convenient for continuous data.
Inclusive form: 1–10, 11–20, ... means both limits are included. To draw a histogram from inclusive data, first convert by subtracting 0.5 from each lower limit and adding 0.5 to each upper limit.

3. Bar Graphs vs Histograms

Both use rectangular bars and look similar at first glance, but represent fundamentally different data types with one critical visual difference.

Why no gaps in a histogram? Class intervals are continuous — 10–20 ends exactly where 20–30 begins. There is no gap in the data scale, so there must be no gap between the bars.

4. Frequency Polygon

A frequency polygon is a line graph that conveys the same information as a histogram but uses connected points instead of bars. Its main advantage: two frequency polygons can be drawn on the same axes to compare two distributions.

Method 1 — From a histogram

1. Mark the class mark (mid-point) at the top centre of each bar.
2. Join consecutive mid-points with straight line segments.
3. Extend the line to the mid-point of a "phantom" class with zero frequency before the first class and after the last class, touching the x-axis. This closes the polygon.

Method 2 — Direct (without histogram)

1. Compute the class mark of each class interval.
2. Plot the points (class mark, frequency) for each class.
3. Also plot (mid-point of phantom class before first, 0) and (mid-point of phantom class after last, 0).
4. Join all points in order with straight line segments.

5. Mean for Ungrouped Data

The arithmetic mean (usually called "the mean" or "average") is found by summing all observations and dividing by the total count.

Here $x_1, x_2, \dots, x_n$ are the $n$ observations and $\bar{x}$ (read "x-bar") is their mean.

Key properties of the mean:

• Unique — every dataset has exactly one mean.
• Uses every observation — the most "information-rich" measure.
• Sensitive to extreme values (outliers). A single very large value can pull the mean far above the "typical" value.
• The sum of deviations from the mean is always zero: $\displaystyle\sum_{i=1}^{n}(x_i - \bar{x}) = 0.$

6. Mean for Grouped Data — Direct Method and Assumed Mean Method

When data is presented in a grouped frequency distribution, individual values are not known. We use the class mark as the representative value for all observations in that class.

Direct Method

where $x_i$ is the class mark of the $i$-th class and $f_i$ is its frequency. Multiply each class mark by its frequency, add up all products, then divide by the total frequency.

Assumed Mean Method (Shortcut)

When class marks are large numbers, the direct multiplications $f_i x_i$ become tedious. Choose an assumed mean $a$ (typically the class mark of the middle class or the class with the highest frequency). Calculate deviations $d_i = x_i - a$ for each class. Then:

This gives the same result with much simpler arithmetic — the $d_i$ values are small (often negative and positive, cancelling out).

7. Median for Ungrouped Data

The median is the value that sits in the exact middle of the data when arranged in ascending order. It divides the distribution into two equal halves — half the values lie below and half above.

Formula

Why median beats mean when outliers exist: Consider salaries of 5 employees: Rs 10,000; 12,000; 11,000; 13,000; 1,00,000.

Mean $= \dfrac{1,46,000}{5} = \text{Rs }29,200$ — this misrepresents 4 of the 5 employees.

Median (3rd value of sorted data) $= \text{Rs }12,000$ — far more representative of a "typical" salary.

8. Mode for Ungrouped Data

The mode is the observation that appears most frequently in the dataset — the value with the highest frequency.

• Unimodal: One value appears most often (most common case).
• Bimodal: Two values tie for most frequent.
• Multimodal: More than two values tie for most frequent.
• No mode: All values appear equally often.

Mode for grouped data (Class 9 introduction): The class interval with the highest frequency is the modal class. The exact formula to find the precise mode within a class is studied in Class 10.

9. Choosing Which Measure to Use

Mean, median and mode each answer a subtly different question about "what is the central value?" Choosing the wrong one gives a misleading picture.

Empirical relationship (for moderately skewed distributions):

This is an approximate empirical formula, not a definition. It is useful in CBSE problems when one of the three measures is unknown and the other two are given.

Summary comparison

• Mean: Unique; uses all values; affected by outliers; need not equal any actual observation.
• Median: Unique; unaffected by outliers; equals an actual observation (odd $n$) or average of two (even $n$).
• Mode: May not be unique; unaffected by extreme values; always an actual observation in the dataset.

10. Common Mistakes to Avoid

• Forgetting to sort data before finding the median — the position formula only works on ordered data.
• Leaving gaps between histogram bars — histograms show continuous data; bars must be adjacent with no spaces.
• Using class limits instead of class marks in the mean formula for grouped data — always compute $x_i = \dfrac{\text{lower} + \text{upper}}{2}$ first.
• Applying HCF-LCM product rule to three numbers (unrelated but a common slip in exam pressure) — stay focused on the chapter's scope.
• Wrong median position for even n — for $n = 10$ the median is the average of the 5th and 6th values, NOT just the 5th.
• Not closing the frequency polygon — always bring the line to the x-axis using zero-frequency anchor points at both ends.
• Computing the mean for qualitative (categorical) data — never take the mean of blood groups, colours, or other non-numeric categories; use mode.
• Confusing bar graph and histogram — bar graphs have gaps (discrete categories), histograms have no gaps (continuous intervals).

11. Quick Revision Checklist

• Primary data = collected by investigator; Secondary data = from existing sources.
• Class mark $= \dfrac{\text{lower} + \text{upper}}{2}$; Class width $= \text{upper} - \text{lower}$.
• Histogram: no gaps, continuous data, bars touch each other.
• Bar graph: gaps between bars, discrete or categorical data.
• Frequency polygon: join class marks at heights = frequencies; close with zero-frequency anchor points.
• Mean (ungrouped) $= \dfrac{\sum x_i}{n}$; Mean (grouped) $= \dfrac{\sum f_i x_i}{\sum f_i}$.
• Assumed mean shortcut: $\bar{x} = a + \dfrac{\sum f_i d_i}{\sum f_i}$ where $d_i = x_i - a$.
• Median: sort first; odd $n$: middle value; even $n$: average of two middle values.
• Mode: most frequent value; use for categorical data or "most popular size" questions.
• Empirical relation: Mode $\approx 3 \times \text{Median} - 2 \times \text{Mean}$.