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AFCAT Numerical Ability · Formula sheet

Statistics formulas

12 formulas, 1 reference table and 26 common traps for AFCAT Numerical Ability Statistics, grouped by subtopic.

Full notes with worked examples

Data, Class Intervals and Pie Charts

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Data and class intervals

Class width from the range

w=H−Lk,last boundary=L+kww = \dfrac{H - L}{k}, \qquad \text{last boundary} = L + kw

Pie charts and square diagrams

Angle of a sector

θ=vT×360∘,side=vs\theta = \dfrac{v}{T} \times 360^\circ, \qquad \text{side} = \sqrt{\dfrac{v}{s}}

Common traps

Every limit moves by the same half-gap

For the classes 15-24 and 25-34 the actual lower limits are 14.5 and 24.5. Moving only the first one, or moving the upper limits down, mixes limits with boundaries.

Census figures are secondary

Data copied from the census or a published report are secondary, even though someone once collected them first-hand. Primary means the investigator collected them.

Out of 360, not out of 100

A sector of 90° is a quarter of the chart, which is 25%. Reading the angle as a percentage gives 90%.

The side is a square root

With 1 cm² for 10 units, 160 units is an area of 16 cm² and a side of 4 cm. Dividing 160 by 10 and calling 16 the side makes the side four times too long.

A ratio needs no total

When the items are given as a ratio, the angle depends on the ratio alone. A large total printed in the question can be ignored.

The Mean, and How Far the Data Spread

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Mean of a list and of a frequency table

Mean of a frequency table

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

How the mean moves when the data change

Correcting a wrong value

xˉcorrect=nxˉ−a+bn\bar x_{\text{correct}} = \dfrac{n \bar x - a + b}{n}

Combined mean of groups

Combined mean

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

Spread: range, variance and standard deviation

Standard deviation

σ=∑(x−xˉ)2n\sigma = \sqrt{\dfrac{\sum (x - \bar x)^2}{n}}

Common traps

Divide by the total frequency

For the values 1 to 4 with frequencies 3, 5, 8 and 4, divide ∑fx\sum fx by 20, the total frequency. Dividing by 4, the number of different values, gives a mean five times too big.

A class counts at its mid-point

The class 10-20 enters the sum as 15. Using 10 or 20 moves every class the same way and shifts the mean by half a class width.

A wrong value does not change the count

Fixing a miscopied value changes the total only. Divide the corrected total by the same number of observations, not by one fewer.

Adding and multiplying act differently

Adding 3 to every value adds 3 to the mean; multiplying every value by 3 triples it. Applying the adding rule to a multiplication, or the other way round, is the usual slip.

Do not average the averages

A group of 10 with mean 20 and a group of 30 with mean 40 combine to 200+120040=35\tfrac{200 + 1200}{40} = 35, not 30. The larger group pulls the mean its way.

Adding a constant does not change the spread

Adding 10 to every value moves the mean by 10, but every distance from the mean stays the same, so σ\sigma does not change.

Take the square root at the end

A variance of 20 is not the standard deviation. σ=20≈4.47\sigma = \sqrt{20} \approx 4.47.

Median, Mode and Comparing the Averages

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Median and mode of a list

Position of the median

median=(n+12)th value(n odd)\text{median} = \left(\dfrac{n + 1}{2}\right)^{\text{th}} \text{ value} \quad (n \text{ odd})

Median of grouped data

M=l+N2−Ff×hM = l + \dfrac{\tfrac N2 - F}{f} \times h

Mode of grouped data

Z=l+f1−f02f1−f0−f2×hZ = l + \dfrac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h

Geometric mean, and numbers from their averages

Geometric mean

G=x1x2⋯xnnG = \sqrt[n]{x_1 x_2 \cdots x_n}

Mean, median and mode compared

Empirical relation

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

Common traps

Order the data first

The middle of 8, 2, 5 as written is 2. In order the list is 2, 5, 8 and the median is 5.

Equal counts mean no mode

In 3, 3, 5, 5, 8, 8 every value occurs twice, so no value is more frequent than the others and the data have no mode.

F is the count before the class

FF is the cumulative frequency of the classes below the median class. Including the median class's own frequency puts the median below its class.

Grouped data use N over 2

The grouped formula uses N2\tfrac N2, not N+12\tfrac{N + 1}{2}. The n+12\tfrac{n + 1}{2} rule is for a list of single values.

Both neighbours go in the denominator

The denominator is 2f1−f0−f22f_1 - f_0 - f_2. With a modal frequency of 22 and neighbours 14 and 10 it is 20; using f1−f2=12f_1 - f_2 = 12 instead gives the wrong mode.

The tallest class, not the middle one

The modal class is the one with the highest frequency, wherever it sits in the table. The median class is a different class in most tables.

Multiply, do not add

The geometric mean of 2 and 18 is 36=6\sqrt{36} = 6, not their ordinary mean 10.

Adding to every value does not add to the geometric mean

1 and 9 have geometric mean 3. Add 1 to each and 2 and 10 have geometric mean 20≈4.47\sqrt{20} \approx 4.47, not 4.

3 times the median, 2 times the mean

Mode =3= 3 Median −2- 2 Mean. Swapping the 3 and the 2 turns a mean of 42 and a median of 40 into a mode of 46 instead of 36.

The mean is not always above the median

In 2, 10, 12 the mean is 8 and the median 10. The mean is above the median only when the large values stretch further from the middle.

Histograms, Frequency Polygons and Ogives

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The ogive and reading the median

Reading the median

cumulative frequency at M=N2\text{cumulative frequency at } M = \dfrac N2

The graphs of a frequency distribution

GraphHow it is drawnWhat you read from it
HistogramTouching rectangles on the class intervals, each with area proportional to its frequencyThe shape of the data, and the modal class (the tallest bar, when the widths are equal)
Frequency polygonDrawn after the histogram: join the mid-points of the tops of its rectangles with straight linesThe same shape as a line, useful for comparing two distributions on one graph
Frequency curveA smooth curve through the same mid-pointsThe overall pattern, such as whether the data are symmetric
Bar chartSeparate bars with gaps, one for each category or valueA comparison between separate categories
OgiveCumulative frequencies plotted at the class boundaries and joinedThe median, read at half the total (the next concept)

Common traps

Area, not height, follows the frequency

When one class is twice as wide as the others, a rectangle of the same height stands for twice the frequency. Halve its height to keep the area right.

A histogram is not a bar chart

A bar chart's bars have gaps and stand for separate categories. A histogram's bars touch because its classes run on from one another.

Less than goes at the upper boundary

The number of items below 20 is plotted at 20, the top of the class 10-20, not at its start or its mid-point.

Half the total frequency, not half the axis

Read the median where the curve reaches N2\tfrac N2 on the vertical axis. The middle of the horizontal axis is not the median unless the data are symmetric.

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