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CDS Mathematics · Formula sheet

Statistics formulas

12 formulas, 2 reference tables and 14 common traps for CDS Mathematics Statistics, grouped by subtopic.

Full notes with worked examples

Data, Scales and Presentation

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Classes, histograms and polygons

Frequency density

frequency density=class frequencyclass width\text{frequency density} = \dfrac{\text{class frequency}}{\text{class width}}

Kinds of data and scales of measurement

ScaleWhat it allowsExample
NominalNaming onlyBlood group, roll number
OrdinalRankingHotel star rating, class rank
IntervalEqual differencesTemperature in °C
RatioEqual differences and ratiosHeight, income, marks
Each scale allows everything the one above it allows, and one thing more.

Common traps

Ranked is not measured

Star ratings and ranks can be ordered, but a 4-star hotel is not 'twice' a 2-star one. That makes them ordinal, not interval or ratio.

Inclusive and exclusive are easy to swap

'15–19, 20–24' is INCLUSIVE (the upper limit is inside the class). '15–20, 20–25' is EXCLUSIVE. A statement question that names them the other way round is false.

Frequency Tables and Cumulative Frequency

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Cumulative frequency tables

Frequency from a cumulative table

f(a–b)=F(b)−F(a)f(a\text{–}b) = F(b) - F(a)

Mean, median and mode of an x–f table

Mean of a frequency table

xˉ=∑fx∑f\bar x = \dfrac{\sum f x}{\sum f}

Common traps

Percent of the maximum, not marks

'Less than or equal to 50%50\% marks' in a test of 8080 means 4040 marks, not 5050. The two readings usually both land on class boundaries, so both are printed as options.

Sort the x values first

Tables often list xx out of order. The mean does not care, but the median does: build the cumulative column only after putting xx in increasing order.

Properties of the Arithmetic Mean

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Shifting and scaling the data

Linear change

y=ax+b  ⇒  yˉ=axˉ+by = ax + b \;\Rightarrow\; \bar y = a\bar x + b

Deviations from the mean and from other points

Deviations

∑(xi−xˉ)=0,∑(xi−A)=n(xˉ−A)\sum (x_i - \bar x) = 0, \qquad \sum (x_i - A) = n(\bar x - A)

Combined and weighted means

Combined mean

xˉ=n1xˉ1+n2xˉ2n1+n2\bar x = \dfrac{n_1\bar x_1 + n_2\bar x_2}{n_1 + n_2}

Common traps

The added constant is added n times to the total

∑(4xi+1)\sum (4x_i + 1) over 1515 values is 4∑xi+154\sum x_i + 15, not 4∑xi+14\sum x_i + 1.

The zero sum needs no arithmetic

When asked for the sum of deviations from the mean, the answer is 00 whatever the data. Computing the mean first wastes time.

Weight by the counts

The mean of two group means is the combined mean only for equal groups. With 3030 and 1010 values the combined mean sits three-quarters of the way toward the larger group's mean.

The Median of a List

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Finding the median

Median position

n odd: (n+12)th;n even: mean of the n2th and (n2+1)thn \text{ odd: } \left(\tfrac{n + 1}{2}\right)\text{th}; \quad n \text{ even: mean of the } \tfrac n2\text{th and } \left(\tfrac n2 + 1\right)\text{th}

How the median responds to changes

Middle of ten values

median=12(x(5)+x(6))\text{median} = \tfrac12\left(x_{(5)} + x_{(6)}\right)

Common traps

Sort before you pick the middle

The middle of the list as printed is not the median. With nn even, average the two middle values of the SORTED list.

Half the change when two values are averaged

If one of the two middle values rises by 11, the median rises by 12\dfrac12, not 11. The mean of the whole list rises by only 1n\dfrac1n.

Mean, Median and Mode of Grouped Data

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Mean of grouped data and missing frequencies

Grouped mean

xˉ=∑fm∑f,m=class mid-point\bar x = \dfrac{\sum f m}{\sum f}, \quad m = \text{class mid-point}

Median of grouped data

Grouped median

Median=l+N2−cff h\text{Median} = l + \dfrac{\tfrac N2 - cf}{f}\,h

Mode of grouped data

Grouped mode

Mode=l+f1−f02f1−f0−f2 h\text{Mode} = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2}\,h

Common traps

An open last class

When the last class has no upper limit, the mean is only possible by assuming a width. Take the width of the other classes; that is the reading the options are built on.

Use boundaries, not printed limits

With inclusive classes the median class 4545–5353 starts at 44.544.5. Using 4545 shifts the answer by 0.50.5, and that shifted value is usually among the options.

Two times f₁ in the denominator

The denominator is (f1−f0)+(f1−f2)=2f1−f0−f2(f_1 - f_0) + (f_1 - f_2) = 2f_1 - f_0 - f_2. Writing f1−f0−f2f_1 - f_0 - f_2 gives a negative or tiny denominator and a mode outside the class.

Choosing a Measure of Central Tendency

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The empirical relation

Empirical relation

Mode=3 Median−2 Mean\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}

Which average to use

MeasureBest forWeakness
MeanBalanced data, further algebraPulled by extreme values
MedianSkewed data, open-ended classesIgnores the size of the other values
ModeMost common item (sizes, categories)May not be unique
Harmonic meanAveraging rates (km/h, Rs./unit)Only for positive values
Geometric meanGrowth rates, ratiosNeeds positive values
The median is the positional average and the one least affected by extreme observations.

Common traps

Positional does not mean 'the mode'

The median is fixed by rank alone, which is why it is called the positional average. Some books also call the mode positional; if both are offered, the median is the standard answer.

Three median, two mean

It is 3 3\,Median −2 - 2\,Mean, not the other way round. Swapping the coefficients gives a 'mode' outside the range of the other two.

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