Data Handling and Measures of Central Tendency
Data Handling and Measures of Central Tendency
Statistics gives 2 questions in the Part-III paper, and unlike Part-II these go past simple averages into grouped data and the effect of changing an observation. The three measures of central tendency and when each is appropriate is the core of it.
Mean, Median and Mode
| Measure | Definition | Affected by extreme values |
|---|---|---|
| Mean | Sum ÷ number of observations | Yes, strongly |
| Median | Middle value when arranged in order | No |
| Mode | Most frequently occurring value | No |
For the data 2, 3, 3, 5, 12: the mean is 5, the median is 3 and the mode is 3. The single large value pulls the mean away from the rest, which is why the median is preferred for skewed data such as incomes.
For an even number of observations, the median is the average of the two middle values after arranging them in order. Forgetting to arrange the data first is the most frequent slip in the topic.
✗ The median of 7, 2, 9, 4 is 9 | ✓ Arrange first: 2, 4, 7, 9 → median is (4 + 7)/2 = 5.5
Properties of the Mean
- Sum of observations = mean × number of observations. Almost every mean word problem uses this line.
- Adding a constant to every observation adds it to the mean; multiplying by a constant multiplies the mean.
- The sum of deviations from the mean is always zero.
- For a replacement, change in total = number of observations × change in mean.
Example: the mean weight of 20 students is 40 kg. When one student leaves, the mean drops to 39.5. The departing student's weight is 800 − (19 × 39.5) = 800 − 750.5 = 49.5 kg.
Grouped Data and Weighted Mean
Mean of grouped data = Σ(f × x) ÷ Σf
Combined mean = (n₁m₁ + n₂m₂) ÷ (n₁ + n₂)
For grouped data, x is the class mark, that is the midpoint of the class interval. Using the lower limit instead of the midpoint is a common error.
The combined mean is weighted by group size. Two classes averaging 60 and 70 do not combine to 65 unless both have the same number of students, and the plain average is always among the options.
Reading Graphs and Tables
- A bar graph compares separate categories; the bars do not touch.
- A histogram shows continuous grouped data; the bars do touch.
- A pie chart shows parts of a whole; degrees = (value ÷ total) × 360, so one per cent is 3.6 degrees.
- The range is the largest value minus the smallest, and it measures spread, not centre.
The bar-graph and histogram distinction is asked directly and turns entirely on whether the data is categorical or continuous — which is also how you would explain it to a class 8 student.
माध्य के लिए आंकड़ों को पहले क्रम में लगाना अनिवार्य है।
TRE pointer: The replacement question, where one observation is swapped and the mean shifts, is the most common statistics item in a teaching paper, and it is solved in one line by change in total = n × change in mean. The bar-graph-versus-histogram distinction is the second. Because a teacher is expected to know when the median beats the mean, expect a reasoning question on skewed data too. With −1/3 negative marking, always sort the data before reading off a median.
60-Second Recap
- Mean is pulled by extreme values; median and mode are not.
- Arrange data in order before finding the median; with an even count, average the two middle values.
- Sum = mean × number of observations; deviations from the mean sum to zero.
- Change in total = number of observations × change in mean.
- Grouped mean uses class marks, not lower limits; combined mean is weighted by group size.
- Bar graph bars do not touch; histogram bars do; one per cent of a pie chart is 3.6 degrees.
Frequently Asked Questions
When is the median a better measure than the mean?
When the data contains extreme values, since the mean is dragged towards them while the median is not. Income data is the standard example: a few very high incomes lift the mean well above what a typical person earns, so the median describes the group better.
What is the difference between a bar graph and a histogram?
A bar graph displays separate categories and its bars are drawn with gaps between them. A histogram displays continuous grouped data and its bars touch, because the class intervals are adjoining. The distinction is about the nature of the data, not the appearance.
How do I find a missing observation when the mean changes?
Use that the change in the total equals the number of observations multiplied by the change in the mean. Compute the original total, compute the new total, and the difference gives the value that was added or removed. This solves the whole family in one line.
What is the class mark in grouped data?
It is the midpoint of a class interval, found by averaging the lower and upper limits. It represents the whole class when computing the mean of grouped data. Using the lower limit instead of the midpoint is the most common error in these calculations.