Statistics — SPI Mathematics Formulas and Traps

Mean of Raw Data

Mean, sum and count

Mean of n values

\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n} \qquad \sum x = n\bar{x}

$\bar{x}$ the mean
$n$ how many values there are

Mean of a number pattern

Means of common patterns

\text{mean of } 1, 2, \dots, n = \frac{n + 1}{2} \qquad \text{mean of first } n \text{ odd numbers} = n \qquad \text{mean of first } n \text{ multiples of } k = \frac{k(n + 1)}{2}

When every value changes the same way

Changing every value

\text{new values } kx + c \;\Rightarrow\; \text{new mean} = k\bar{x} + c \qquad \sum (x - \bar{x}) = 0

Common traps

Dividing by the number of known values

In a list like 5, x, 9, y, 11 there are five values, so the sum is 5 times the mean. Leaving the unknowns out of the count gives a wrong sum.

Looking for x and y separately

One mean gives one equation. With two unknowns it fixes only their sum, so the right option is the one that states

x + y

, not a value of x alone.

Starting whole numbers at 1

The first six whole numbers are 0 to 5, not 1 to 6. Starting at 1 gives the answer for natural numbers, and that answer is usually printed as a wrong option.

Taking the sum for the mean

The first n odd numbers add to

n^2

, but their mean is

n^2 \div n = n

. Divide by the count before you compare with the options.

Using first plus last over two for squares

The first and last trick works only when the numbers go up by a fixed step. The squares 1, 4, 9, 16 do not, so add them up instead.

Adding n times the change to the mean

Adding 3 to each of n values adds

3n

to the total, but the count is also n, so the mean rises by 3. 'Increases by 3n' is a printed wrong option.

Doing the steps in the wrong order

If the mean is m, dividing every value by x and then adding y gives a new mean of

\frac{m}{x} + y

. Adding first and then dividing gives

\frac{m + y}{x}

, which is a different number.

Median and Mode of Raw Data

Learn this subtopic in the notes

Median of a list

Position of the median

n \text{ odd: } \left(\frac{n + 1}{2}\right)\text{th value} \qquad n \text{ even: mean of the } \left(\frac{n}{2}\right)\text{th and } \left(\frac{n}{2} + 1\right)\text{th values}

Mean, median and mode side by side

Measure	What it is	How to find it
Mean	The average	Add all the values and divide by how many there are
Median	The middle value	Put the values in order and take the middle one
Mode	The most frequent value	Count how often each value occurs
Smallest and largest value removed	The median stays the same	The middle of an ordered list is still the middle The mean uses every value, so it usually changes. The mode changes only if an end value was the most frequent one.

The mode is the most frequent value, the median the middle one: 'middle most frequent' is a mix of the two.

Common traps

Calling the mode the middle value

The middle value of ordered data is the median. The mode is the most frequent value, whether or not it sits in the middle.

Taking the middle of an unordered list

In 59, 46, 30, 23, 27 the middle number as written is 30, but in order the list is 23, 27, 30, 46, 59. Always sort before you count to the middle.

Giving the position for the value

For 9 values,

\frac{9 + 1}{2} = 5

says the median is the 5th value. The answer is whatever number stands 5th, not 5.

Losing count in a long list

In a list of 15 numbers, two values can look equally common at a glance. Tally them and check that the tallies add to 15 before you choose.

Mean of Grouped Data

Learn this subtopic in the notes

Class marks and the direct method

Direct method

\text{class mark} = \frac{\text{lower} + \text{upper}}{2} \qquad \bar{x} = \frac{\sum f_i x_i}{\sum f_i}

$f_i$ frequency of a class
$x_i$ class mark of that class

Assumed mean method

d_i = x_i - a \qquad \bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}

$a$ assumed mean
$d_i$ deviation of a class mark from a

Step deviation method

u_i = \frac{x_i - a}{h} \qquad \bar{x} = a + h \times \frac{\sum f_i u_i}{\sum f_i}

$h$ class size, the same for every class

Common traps

Multiplying by a class limit

Each frequency is multiplied by the class mark, the mid-point. Using the lower or upper limit shifts the mean by half a class.

Dividing by the number of classes

The mean of a table is

\sum f_i x_i

over the total frequency

\sum f_i

. A table with 6 classes and 50 students divides by 50, not 6.

Using only one equation for two unknown frequencies

When two frequencies are missing, the total frequency gives one equation and the mean gives the other. Solve them together, then check that the frequencies add to the total.

Dropping the minus signs

Class marks below the assumed mean give negative deviations. Adding them as positive numbers makes

\sum f_i d_i

too large and the mean too high.

Forgetting to multiply by h

In the step deviation method,

\frac{\sum f_i u_i}{\sum f_i}

counts steps, not marks. Multiply it by the class size h before adding it to a.

Median and Mode of Grouped Data

Learn this subtopic in the notes

Cumulative frequency and the median class

Median class

\text{median class: first class with } cf \ge \frac{N}{2}

The median formula

Median of grouped data

\text{Median} = l + \frac{\frac{N}{2} - cf}{f} \times h

$l$ lower limit of the median class
$N$ total frequency
$cf$ cumulative frequency of the class before the median class
$f$ frequency of the median class
$h$ class size

The mode formula

Mode of grouped data

\text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h

$l$ lower limit of the modal class
$f_1$ frequency of the modal class
$f_0$ frequency of the class before it
$f_2$ frequency of the class after it
$h$ class size

Common traps

Picking the tallest class as the median class

The class with the largest frequency is the modal class. The median class is found from the cumulative frequencies and is often a different class.

Using the median class's own cumulative frequency

cf in the median formula is the running total up to the class before the median class. Using the median class's own cumulative frequency makes

\frac{N}{2} - cf

negative or too small.

Plugging mid-values in as limits

When a table lists mid-values, the lower limit l is the mid-value minus half the class size. A median class with mid-value 45 and size 10 has l = 40, not 45.

Swapping f0 and f1 in the top line

The top of the mode fraction is

f_1 - f_0

, the modal class minus the class before it. Writing

f_0 - f_1

makes it negative and puts the mode below the modal class.

Using classes with gaps as they are

For classes like 41-45 and 46-50, the true limits are 40.5-45.5 and 45.5-50.5. Using 46 as l without the 0.5 shift gives the wrong mode.

Empirical Relation and Graphs

Learn this subtopic in the notes

Mode equals 3 median minus 2 mean

Empirical relation

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

Which graph shows what

Graph	What it shows	What you can read from it
Histogram	Class intervals on the x-axis, frequencies on the y-axis, bars touching	The mode, from the tallest bar
Bar graph	Separate items as bars with gaps between them	Which item is largest or smallest
Pie chart	A whole divided into sectors for its parts	The share of each part: sector angle = part ÷ whole × 360°
Pictograph	Quantities drawn as rows of pictures	Counts, from the number of pictures
Frequency polygon	The mid-points of the histogram bar tops joined by lines	The shape of the distribution
Ogive	Cumulative frequencies plotted against class limits	The median, where the less-than and more-than ogives cross No graph gives the mean: it must be calculated.

The mean is the one average that cannot be found from a graph.

Common traps

Swapping mean and median

The 3 goes with the median and the 2 with the mean: Mode

= 3

Median

- 2

Mean. 'Mode

= 3

Mean

- 2

Median' is the printed wrong option.

Losing the minus sign when matching a pattern

Written as x Median

+

y Mean

= z

Mode, the relation has

x = 3

,

y = -2

and

z = 1

. The minus sign belongs to y.

Putting the classes on the y-axis

In a histogram the class intervals run along the x-axis and the frequencies up the y-axis. A histogram's bars also touch, unlike a bar graph's.

Thinking every average has a graph

The median comes from an ogive and the mode from a histogram, but the mean has no graphical method. It is the answer when a question asks which cannot be found graphically.

Statistics formulas

Mean of Raw Data

Mean, sum and count

Mean of a number pattern

When every value changes the same way

Common traps

Median and Mode of Raw Data

Median of a list

Mean, median and mode side by side

Common traps

Mean of Grouped Data

Class marks and the direct method

Common traps

Median and Mode of Grouped Data

Cumulative frequency and the median class

The median formula

The mode formula

Common traps

Empirical Relation and Graphs

Mode equals 3 median minus 2 mean

Which graph shows what

Common traps

More SPI Mathematics formula sheets