Statistics 1: Data, Mean, Median and Mode

SSC · RRB · State exams — method by method

6 Part9 worked examples18 practice questions

Four students scored 30, 40, 40 and 50. Instead of remembering the whole list we want to describe it by one number: “about 40 marks”. Statistics does exactly this: it collects data, arranges it, summarises it in one or two numbers and draws conclusions from it.

One number is not enough. Marks of 40, 40, 40, 40 or of 10, 20, 60, 70 both have mean 40, yet the two classes are very different. So two questions are asked: (1) where is the centre of the data? and (2) how widely is it spread around the centre?

Question	Measures	In this chapter
Where is the centre?	mean, median, mode	this chapter: groups B, C, D
How spread out?	range, mean deviation, variance, standard deviation, coefficient of variation	Statistics 2
How many are below a point? What is the shape?	quartiles, percentiles, skewness, the normal curve	Statistics 2

Here: the first group (A) gives the language of tables and pictures; group E (reading tables) holds the table-reading skills and the exam traps. The second and third questions (spread and shape) are answered fully in the next chapter, Statistics 2: Spread, Quartiles and Distribution Shape.

LinksOur Average chapter covers the total-and-count questions on the mean (two groups, one value joining or leaving and so on) in detail, and the Data Interpretation chapters cover reading charts. Here we build the structure of statistics on top of them: frequency tables, the mean of grouped data, the median and the mode.

Data and frequency tables

Tables · classes · pictures

Kinds of data, tally and frequency, class intervals and limits, and showing data in pictures.

Before any measure can be found the data must be arranged. These three parts are the language of every later method: what a data value is and how it is counted, every term of a class interval, and how data is shown in pictures.

Data, variable and frequency

Tally · frequency

Kinds of data, raw and grouped data, tally marks, frequency and relative frequency.

The idea

Each single value that we count or measure (a mark, an age, a height, the number of children in a family) is an observation or a data value, and the thing being measured is the variable. A list as first collected is called raw data. When the same value comes again and again it is easier to count and write it once; the number of times a value occurs is its frequency, and such a list is a frequency table.

Term	Meaning	Example
Discrete variable	takes separate, countable values	children in a family, goals, shoe size
Continuous variable	any value between two values is possible	height, weight, time, rainfall
Raw data	the original list, unordered and ungrouped	12, 15, 12, 18, 15, 12
Tally mark	a counting stroke; the fifth stroke crosses the four	\|\|\|\|
Frequency $f$	how many times a value occurs	12 → 3 times
Relative frequency	frequency ÷ total (also as a percentage)	$\frac{3}{6} = 50%$

Here: $f$ = frequency, $N$ = total frequency ( $N = \sum f$ , the sum of all frequencies).

∑ Formula

N = \sum f, relative frequency = \frac{f}{N}

Total and relative frequency

$f$: frequency of one value or class
$N$: total frequency = total number of observations
$\sum f$: the sum of all the frequencies (Σ means “sum of”)

CheckThe relative frequencies always add up to 1 (or 100%). That is the check that no value was missed or counted twice.

Steps: from raw data to a table

Write all the different values, smallest to largest, in one column.
Read the list once from top to bottom and put a tally mark against each value.
Count the marks to get the frequency $f$ and check that $\sum f$ equals the number of observations.
If needed add a column of relative frequency $\frac{f}{N}$ and of cumulative frequency (the running total).

Example 1 · Basic

The number of goals scored in each of 24 matches of a league was recorded: 5, 3, 3, 4, 3, 3, 0, 2, 1, 6, 1, 2, 2, 1, 4, 4, 5, 4, 1, 3, 6, 6, 5, 5. What is the frequency of the value 2?

Solution, step by step

1Read the list once, put a tally mark for each value and count. Counts: 0→1, 1→4, 2→3, 3→5, 4→4, 5→4, 6→3. Their sum is $24$ , so no value was missed or counted twice.
2The count against 2 is 3; that is its frequency.

Answer 3

Strike each value off the list as you count it.

Example 2 · Standard

At a bus stand the number of empty seats (from 1 to 5) in each of 20 buses was noted: 3, 3, 1, 4, 5, 1, 2, 1, 4, 1, 5, 3, 2, 4, 3, 1, 1, 5, 1, 5. Make a table with frequency, relative frequency (%) and cumulative frequency.

Solution, step by step

1The values run from 1 to 5, so make five rows, read the list once while putting tally marks, and count: 1→7, 2→2, 3→4, 4→3, 5→4.
2Check: $7 + 2 + 4 + 3 + 4 = 20$ , so all 20 buses were counted.
3Relative frequency $\frac{f}{20} \times 100$ : 35%, 10%, 20%, 15%, 20%; they add up to 100%.
4Cumulative frequency (running total): 7, 9, 13, 16, 20; the last number 20 equals the total 20.

Answer $Empty seats Frequency Cumulative frequency (c f) 177229341343165420$

The last cumulative frequency must equal the total; check it.

Example 3 · Standard

The number of broken biscuits in each of 24 packets was recorded: 0, 1, 3, 3, 2, 2, 2, 0, 2, 4, 0, 3, 4, 2, 2, 3, 2, 3, 4, 4, 2, 2, 3, 1. What is the relative frequency of the value 3, as a percentage?

Solution, step by step

1Read the list once, put a tally mark for each value and count. Counts: 0→3, 1→2, 2→9, 3→6, 4→4. Their sum is $24$ , so no value was missed or counted twice.
2Relative frequency $= \frac{frequency}{total} = \frac{6}{24}$ .
3As a percentage: $\frac{6}{24} \times 100 = 25%$ .

Answer 25%

The denominator is the count of all the values, not of that value.

Practice

BasicTry it yourself first

The number of members in each of 24 families was recorded: 6, 6, 2, 3, 6, 2, 4, 4, 8, 4, 3, 4, 5, 8, 6, 3, 5, 6, 6, 4, 5, 4, 5, 4. What is the frequency of the value 6?

BasicTry it yourself first

The number of books borrowed in a week by each of 24 students from a library was recorded: 2, 4, 4, 4, 2, 6, 3, 1, 3, 4, 2, 5, 3, 3, 5, 3, 5, 4, 3, 5, 1, 7, 3, 4. What is the frequency of the value 6?

StandardTry it yourself first

The number of customers who came to a shop on each of 24 days was recorded: 15, 25, 20, 30, 20, 15, 25, 35, 15, 20, 20, 20, 20, 25, 25, 20, 20, 15, 20, 20, 15, 35, 25, 15. What is the relative frequency of the value 20, as a percentage?

StandardTry it yourself first

The number of passengers in each of 24 buses was recorded: 40, 30, 50, 40, 30, 70, 50, 60, 60, 30, 40, 60, 60, 20, 50, 50, 40, 40, 40, 50, 50, 30, 30, 50. What is the relative frequency of the value 30, as a percentage?

BasicTry it yourself first

Which of these is a continuous variable?

StandardTry it yourself first

In a frequency table of 30 observations one value has relative frequency 0.2. What is its frequency?

Class intervals, limits and class marks

The language of grouped data

Class interval, lower and upper limits, true boundaries, class width, class mark, cumulative frequency (less than and more than).

The idea

When there are many values or the variable is continuous (like the marks of 30 students) it is useless to write the frequency of every value. Instead the values are put into fixed ranges called class intervals, such as 0–10, 10–20, 20–30, and we count how many values fall in each range. This is grouped data. A class has two ends: the lower limit and the upper limit.

Term	Meaning	In the class 20–30
Lower / upper limit	the two ends of the class	20 / 30
Class width $h$	upper − lower limit	$30 - 20 = 10$
Class mark $x$	the middle value: average of the limits	$\frac{20 + 30}{2} = 25$
Cumulative frequency (cf)	running total of the frequencies up to this class	sum for 0–10, 10–20, 20–30

Here: $h$ and $x$ appear in every later formula; in grouped data all the values of a class are treated as if they were at its class mark $x$ .

Two kinds of classes: exclusive and inclusive

In exclusive classes 0–10, 10–20, 20–30 the upper limit of one class is the lower limit of the next. The rule: a boundary value is counted in the next class, so a student with 10 marks is in 10–20, not in 0–10. In inclusive classes 20–29, 30–39, 40–49 both limits belong to the same class and there is an apparent gap from 29 to 30. For calculations (median and so on) this gap must be closed: add 0.5 to the upper limit and subtract 0.5 from the lower limit to get the true boundaries.

◭ Diagram

The true boundaries of the inclusive classes 20–29 and 30–39 are 19.5, 29.5, 39.5; the gap of 1 is shared half and half

∑ Formula

x = \frac{lower + upper}{2}, h = upper - lower, boundary = limit \mp \frac{gap}{2}

Class mark, width, true boundary

$x$: class mark (middle of the class)
$h$: class width (difference of the true boundaries, or of two consecutive lower limits)
$gap$: difference between a class’s upper limit and the next class’s lower limit (1 for inclusive classes)

TrapAlways take the class width as the difference of two consecutive lower limits: for 20–29, 30–39 it is

30 - 20 = 10

, not

29 - 20 = 9

. This is the commonest trap.

Steps: from raw data to a grouped table

Find the range (largest − smallest) and choose a class width (a convenient width giving 5 to 10 classes: 5, 10, 20 …).
Number of classes $= \frac{range}{width}$ , rounded up if it is a fraction.
Write the classes (the first starts at or below the smallest value), then put a tally mark for each value; a value on a boundary goes into the next class.
Count the frequencies; check $\sum f$ against the number of observations; add cf if needed.

Example 4 · Basic

The marks (out of 50) of 30 students are: 19, 16, 21, 39, 8, 32, 22, 46, 33, 11, 46, 21, 15, 11, 14, 13, 34, 24, 37, 9, 38, 30, 20, 33, 18, 4, 18, 33, 5, 11. Make a frequency table with class width 10 and the cumulative frequency (less than).

Solution, step by step

1The smallest is 4 and the largest 46, so five classes cover 0 to 50: 0–10, 10–20, 20–30, 30–40, 40–50 (width 10, 5 classes).
2A tally mark goes in the right class for every mark; a mark on a boundary (such as 10, 20, 30, 40) goes into the next class. Counts: 4, 10, 5, 9, 2.
3Check: $4 + 10 + 5 + 9 + 2 = 30$ , so all 30 students were counted.
4“Less than” cf: below 10 → 4; below 20 → 14; below 30 → 19; below 40 → 28; below 50 → 30.

Answer Frequencies 4, 10, 5, 9, 2; cf 4, 14, 19, 28, 30

For a “more than” cf subtract the previous cf from the total.

Example 5 · Basic

What is the class mark of the class 80–90?

Solution, step by step

1The class mark is the middle value of the class: the average of its lower and upper limits.
2 $\frac{80 + 90}{2} = 85$ .
3Check: 85 is 5 above the lower limit 80 and 5 below the upper limit 90, so it is exactly in the middle.

Answer 85

The class mark always lies between the two limits.

Example 6 · Standard

What are the true class boundaries of the inclusive class 70–79 (both limits included)?

Solution, step by step

1The next class is 80–89, so the gap between 79 and 80 is 1. To close it we add $0.5$ to the upper limit and subtract $0.5$ from the lower limit.
2True boundaries: $70 - 0.5 = 69.5$ and $79 + 0.5 = 79.5$ .
3Check: the width of the true boundaries is $79.5 - 69.5 = 10$ , equal to the difference of consecutive lower limits $80 - 70 = 10$ ; so the next class starts exactly at 79.5 and no gap is left.

Answer 69.5–79.5

Half the gap: here

\frac{1}{2} = 0.5

Practice

BasicTry it yourself first

What is the class mark of the class 100–120?

StandardTry it yourself first

The class marks of consecutive classes in a frequency table are 57.5, 62.5, 67.5, 72.5. What are the limits of the first class?

StandardTry it yourself first

The smallest value of a data set is 60 and the largest is 91. How many classes of width 5 are needed to cover it?

StandardTry it yourself first

What are the true class boundaries of the inclusive class 70–79 (both limits included)?

StandardTry it yourself first

The delivery times of 100 orders on a food-delivery app are given below:

Class interval Frequency (f) 10 - 15 11 15 - 20 18 20 - 25 22 25 - 30 20 30 - 35 18 35 - 40 11

. How many have a value less than 25?

StandardTry it yourself first

The ages of 80 patients who visited a clinic are given below:

Class interval Frequency (f) 20 - 30 11 30 - 40 15 40 - 50 17 50 - 60 18 60 - 70 10 70 - 80 9

. How many have a value of 60 or more?

Showing data in pictures

Histogram · polygon · ogive · pie

Bar graph, histogram, frequency polygon, cumulative frequency curve (ogive), pie chart and unequal class widths.

The idea

A picture shows the shape of the data at a glance: where it piles up and where it is empty. In a histogram each class is a rectangle; the rectangles touch because there is no gap between class intervals (a bar graph for separate categories leaves gaps between the bars). The height of a rectangle is the frequency.

◭ Diagram

Blue: histogram (the tallest rectangle in orange: the modal class); pink: the frequency polygon, joining points above the class marks and reaching the empty imaginary classes at both ends

Picture	How it is drawn	Used for
Bar graph	separate bars with gaps; height = value	comparing categories
Histogram	touching rectangles; width = class, height = frequency	shape of continuous data
Frequency polygon	points at the class marks joined by straight lines, plus an empty class at each end	comparing two data sets
Ogive (cumulative curve)	“less than”: cf at the upper boundary; “more than”: cf at the lower boundary	median and percentiles
Pie chart	angle of each part = $\frac{f}{N} \times 36 0^{\circ}$	share of each part

∑ Formula

θ = \frac{f}{N} \times 36 0^{\circ}, height = \frac{f}{w} \times h_{0}

Pie angle and the height for unequal widths

$θ$: angle of that sector of the pie
$f$: frequency of that class
$N$: total frequency
$w$: width of that class
$h_{0}$: the standard class width (the smallest or usual width)

TrapWith unequal widths the area must be proportional to the frequency, so lower the height of a wide class. Taking the class with the biggest frequency as the modal class from the height alone is a mistake.

Steps

Histogram: true class boundaries on the horizontal axis, frequency on the vertical; a rectangle of that height over each class.
Polygon: a point at the middle (class mark) of the top side of each rectangle; join the points and extend to the imaginary empty classes at both ends.
Ogive: plot the points (upper boundary, cf) and begin with (lower boundary, 0).
Pie: find each angle $\frac{f}{N} \times 36 0^{\circ}$ ; the angles must add up to $36 0^{\circ}$ .

◭ Diagram

Total 50: the class 90–100 (frequency 5) has angle

\frac{5}{50} \times 36 0^{\circ} = 3 6^{\circ}

Example 7 · Basic

The distances covered in a day by 40 delivery riders are to be shown in a pie chart:

Class interval Frequency (f) 0 - 5 5 5 - 10 9 10 - 15 10 15 - 20 9 20 - 25 7

. What is the angle of the sector for the class 0–5?

Solution, step by step

1The full circle of $36 0^{\circ}$ stands for the total 40.
2Angle of this class $= \frac{5}{40} \times 36 0^{\circ} = 4 5^{\circ}$ .
3Check: the angles of all the classes are 45°, 81°, 90°, 81°, 63°; they add up to $36 0^{\circ}$ , the full circle.

Answer 45°

Angle = (frequency ÷ total) × 360°.

Example 8 · Standard

The monthly electricity use of 100 houses of a colony are given below:

Class interval Frequency (f) 100 - 120 13 120 - 140 19 140 - 160 23 160 - 180 20 180 - 200 13 200 - 220 12

. How many have a value of 140 or more?

Solution, step by step

1“140 or more” are those not below 140. First the number below 140: $13 + 19 = 32$ .
2The rest $= 100 - 32 = 68$ .
3Check: “below 140” (32) and “140 or more” (68) together make $32 + 68 = 100$ , the whole total; no value is missed or counted twice.

Answer 68

“More” = total − “less”.

Example 9 · Advanced

A frequency table with unequal widths is to be drawn as a histogram:

Class interval Frequency (f) 140 - 145 4 145 - 150 6 150 - 160 16 160 - 170 20 170 - 185 36

. What height (adjusted frequency) should the bar of the class 170–185 have, taking a width of 5 as the standard?

Solution, step by step

1The width of this class is $185 - 170 = 15$ , while the standard width is 5.
2With unequal widths the frequency must be scaled by the ratio of the widths: $36 \times \frac{5}{15} = 12$ .
3Check: this class is wider than the standard width 5, so its bar must be lower than its frequency 36: 12 < 36. Area check: 12 $\times$ 3 (the number of 5-wide parts of the class) $= 36$ , which equals the frequency.

Answer 12

Lower a wide bar so that area stays proportional to frequency.

Practice

BasicTry it yourself first

The daily wages of 72 workers of a factory are to be shown in a pie chart:

Class interval Frequency (f) 300 - 350 9 350 - 400 14 400 - 450 19 450 - 500 18 500 - 550 12

. What is the angle of the sector for the class 450–500?

BasicTry it yourself first

The heights of 60 students of a school are to be shown in a pie chart:

Class interval Frequency (f) 140 - 145 11 145 - 150 13 150 - 155 14 155 - 160 16 160 - 165 6

. What is the angle of the sector for the class 160–165?

StandardTry it yourself first

A frequency table with unequal widths is to be drawn as a histogram:

Class interval Frequency (f) 0 - 50 10 50 - 100 15 100 - 200 40 200 - 300 50 300 - 450 90

. What height (adjusted frequency) should the bar of the class 300–450 have, taking a width of 50 as the standard?

StandardTry it yourself first

A frequency table with unequal widths is to be drawn as a histogram:

Class interval Frequency (f) 100 - 120 6 120 - 140 9 140 - 180 24 180 - 220 30 220 - 280 54

. What height (adjusted frequency) should the bar of the class 140–180 have, taking a width of 20 as the standard?

StandardTry it yourself first

The heights of 50 students of a school are given below:

Class interval Frequency (f) 140 - 145 6 145 - 150 8 150 - 155 10 155 - 160 10 160 - 165 8 165 - 170 8

. What is the relative frequency of the class 155–160, as a percentage?

StandardTry it yourself first

In a “less than” ogive, at which x-value is the cumulative frequency of the class 30–40 plotted?

The mean (arithmetic average)

Total ÷ count · in every form

The mean of raw, tabulated and grouped data, weighted and combined means, one operation on all values, corrections, and series.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Mean: a list and a frequency table

\overset{x}{ˉ} = \frac{\sum x}{n}

\frac{\sum f x}{\sum f}

The mean of a list, the mean of a frequency table and the properties of the mean.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Mean: grouped data (three methods)

Direct · assumed mean · step deviation

The mean from class marks: the direct method, the assumed-mean method and the step-deviation method.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Weighted mean and combined mean

Every value has its own weight

The mean of values with weights, the combined mean of two groups and the ratio of the groups.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

The same operation on every value

Add · multiply · percent

The effect on the mean of adding, subtracting, multiplying or raising by a percent every value, and the reverse question.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

One value joins, leaves, changes or was wrong

Change the total, get the new mean

A value joining, leaving or being replaced, and the correction of a wrongly noted value.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Missing frequencies and missing values

Build an equation from the mean

Finding one or two missing frequencies of a table, or missing values of a list, from the given mean.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

The mean of a series

Consecutive · even · odd · squares · multiples

The mean of natural, even and odd numbers, squares, multiples and an equally spaced series.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

The median

The middle value · the cf formula

The median of raw data and of grouped data (from cumulative frequency), missing frequencies, and the median from an ogive.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Median: raw data

Arrange · take the middle

The middle value, or the average of the two middle values, after arranging; and the steadiness of the median.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Median: grouped data

M = L + \frac{N /2 - c f}{f} \times h

The median class and its formula, the median from a “less than” table, the ogive, and missing frequencies.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

The median and counts from an ogive

Where the two curves cross = the median

The “less than” and “more than” ogives, the median from their crossing point, and the number of values below or between given points.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

The mode

The most frequent

The mode of raw and grouped data, the empirical relation, and choosing the right average.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Mode: raw data

Count · take the highest

Finding the mode, bimodal data (two modes) and data with no mode.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Mode: grouped data

Mo = L + \frac{f _{1} - f _{0}}{2 f _{1} - f _{0} - f _{2}} \times h

Identifying the modal class and the mode formula; caution with unequal widths.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Mean, median, mode: the relation and the right average

= 3

- 2

Mean

The empirical relation, the order of the three, and which average to choose when.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Solving from tables and charts

Reading · traps · mixed

Reading answers straight from a frequency table, and the common exam traps.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Reading a frequency table

Commonest class · percent · ratio

The modal class, the percent “or more”, the ratio and share of classes, and cumulative counts.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Traps: inclusive classes, unequal widths, open ends

Where marks are lost

The true boundaries and median of inclusive classes, the modal class for unequal widths, and open-ended classes.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Summary, mixed practice, revision

All formulas in one place

One table of all the formulas of this chapter, over 40 exam-style mixed questions and revision cards.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Summary of all formulas

The whole chapter on one page

The formulas for the language of data, the mean, median and mode, and table reading, each with one thing to remember.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Mixed practice — set 1

Basic to standard

Questions from all methods mixed; spot the method first, then solve.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Mixed practice — set 2

Standard to advanced

Mixed questions from the hard end of the exam.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in

Revision cards and exam tips

Flip, then recall

Flip cards of the most important facts, last tips, and the way to the next chapter.

🔒

This part is in All Access.

With the pass, every example, practice question and solution in this chapter opens.

See the pass · ₹100.00 Already have a pass? Sign in