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EAPCET Mathematics · Intermediate second year · Paper IIA

Measures of Dispersion

Measures of Dispersion describes how spread out data is, through range, mean deviation, variance, standard deviation and the coefficient of variation.

Subject
Mathematics
Taught in
Intermediate second year · Paper IIA
Mathematics in the paper
80 of 160 questions

Key formulas and results

  • Variance σ2=∑xi2n−xˉ2\sigma^{2} = \dfrac{\sum x_{i}^{2}}{n} - \bar{x}^{2}; standard deviation σ=variance\sigma = \sqrt{\text{variance}}
  • Mean deviation about the mean =∑∣xi−xˉ∣n= \dfrac{\sum |x_{i} - \bar{x}|}{n}
  • Coefficient of variation =σxˉ×100= \dfrac{\sigma}{\bar{x}} \times 100
  • Adding a constant to every value leaves the variance unchanged; multiplying by kk multiplies σ\sigma by ∣k∣|k|
  • Variance of the first nn natural numbers =n2−112= \dfrac{n^{2} - 1}{12}
  • Mean deviation about the median =∑fi∣xi−M∣∑fi= \dfrac{\sum f_{i}|x_{i} - M|}{\sum f_{i}}
  • Median of grouped data =l+(N/2−cff)×h= l + \left(\dfrac{N/2 - cf}{f}\right) \times h
  • Mean by step deviation: xˉ=A+(∑fidi∑fi)×h\bar{x} = A + \left(\dfrac{\sum f_{i}d_{i}}{\sum f_{i}}\right) \times h, where di=xi−Ahd_{i} = \dfrac{x_{i} - A}{h}
  • Combined mean of two groups: xˉ=n1xˉ1+n2xˉ2n1+n2\bar{x} = \dfrac{n_{1}\bar{x}_{1} + n_{2}\bar{x}_{2}}{n_{1} + n_{2}}
  • Combined variance: σ2=n1(σ12+d12)+n2(σ22+d22)n1+n2\sigma^{2} = \dfrac{n_{1}(\sigma_{1}^{2} + d_{1}^{2}) + n_{2}(\sigma_{2}^{2} + d_{2}^{2})}{n_{1} + n_{2}}, where d1=xˉ1−xˉd_{1} = \bar{x}_{1} - \bar{x} and d2=xˉ2−xˉd_{2} = \bar{x}_{2} - \bar{x}
  • Coefficient of range =L−SL+S= \dfrac{L - S}{L + S}, with LL and SS the largest and smallest values

What this chapter covers

  • Range
  • Mean deviation
  • Variance and standard deviation
  • Coefficient of variation
  • Grouped and ungrouped data

How it is usually asked

  • Find the variance or standard deviation of a data set
  • Apply a change of origin or scale
  • Compare consistency using the coefficient of variation

Understanding Measures of Dispersion

Two data sets can have the same mean but be spread very differently, and measures of dispersion describe that spread. The range, largest minus smallest, is quick but depends only on the extremes. The mean deviation averages the absolute distances of the values from the mean or median. The variance averages the squared deviations from the mean, and the standard deviation σ is its square root, expressed in the same units as the data.

For calculation, the variance is σ² = Σxᵢ²/n − x̄², the mean of the squares minus the square of the mean. For grouped data each term is weighted by its frequency. Change of origin and scale simplifies the arithmetic: adding a constant to every value shifts the mean but leaves the variance unchanged, while multiplying every value by k multiplies σ by |k| and the variance by k².

To compare the consistency of two data sets with different means, use the coefficient of variation, (σ/x̄) × 100; the set with the smaller value is more consistent. EAPCET questions ask for variances of small data sets, the effect of transforming data, standard results like the variance (n² − 1)/12 of the first n natural numbers, and consistency comparisons.

Worked example

EAPCET-style question

The variance of 2, 4, 6, 8, 10 is 8. Each observation is multiplied by 3 and then increased by 5. The variance of the new data is

  1. (1)24
  2. (2)29
  3. (3)72
  4. (4)77

Solution

  1. Adding 5 to every value is a change of origin and does not change the variance.
  2. Multiplying every value by 3 multiplies the variance by 32=93^{2} = 9.
  3. New variance =9×8=72= 9 \times 8 = 72.
  4. Check: the new data 11, 17, 23, 29, 35 has mean 23 and squared deviations 144, 36, 0, 36, 144, which average to 72.

Answer: 72

Shortcuts and results to remember

  • Write ungrouped data in increasing order before finding the median.
  • Empirical relation for a moderately skewed distribution: Mode = 3 Median − 2 Mean; for a symmetric one, mean = median = mode.

More practice questions

Practice question 1

Two groups of 10 observations each have means 20 and 30 and the same variance 4. The variance of all 20 observations together is

  1. (1)29
  2. (2)4
  3. (3)25
  4. (4)54

Solution

  1. Combined mean =10×20+10×3020=25= \dfrac{10 \times 20 + 10 \times 30}{20} = 25
  2. d1=20−25=−5d_{1} = 20 - 25 = -5 and d2=30−25=5d_{2} = 30 - 25 = 5
  3. σ2=10(4+25)+10(4+25)20\sigma^{2} = \dfrac{10(4 + 25) + 10(4 + 25)}{20}
  4. =58020=29= \dfrac{580}{20} = 29

Answer: 29

Common mistakes to avoid

  • Changing the variance when a constant is added to every observation; a change of origin leaves both variance and standard deviation unchanged.
  • Multiplying the variance by k when the data is multiplied by k; the variance is multiplied by k², the standard deviation by |k|.
  • Writing σ² = Σxᵢ²/n − x̄ instead of subtracting the square of the mean, x̄².
  • Leaving out the absolute value in the mean deviation; without it the deviations from the mean always add to zero.

How to prepare it

Use change of origin and scale to keep the numbers small — it saves time on grouped data.

Once the formulas feel familiar, test them the way the exam does: practise Measures of Dispersion questions chapter-wise, then attempt a full previous year paper to see whether it holds up under a 3-hour timer.

Related Mathematics chapters

See every chapter in the EAPCET Mathematics syllabus.

Practise Measures of Dispersion chapter-wise

Chapter-wise quizzes show exactly which topics in this chapter you still get wrong — before a full paper does.

Frequently asked questions

Is Measures of Dispersion part of the EAPCET Mathematics syllabus?
Yes. Measures of Dispersion is taught in Intermediate second year · Paper IIA Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.
How should I prepare Measures of Dispersion for EAPCET?
Use change of origin and scale to keep the numbers small — it saves time on grouped data.
How many Mathematics questions are there in EAPCET?
80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.
What mistakes do students make in Measures of Dispersion?
Changing the variance when a constant is added to every observation; a change of origin leaves both variance and standard deviation unchanged. Multiplying the variance by k when the data is multiplied by k; the variance is multiplied by k², the standard deviation by |k|. Writing σ² = Σxᵢ²/n − x̄ instead of subtracting the square of the mean, x̄². Leaving out the absolute value in the mean deviation; without it the deviations from the mean always add to zero.