Making sense of numbers - a half-day workshop

© 2015 Copyright ISC Ltd.
Making sense of numbers
An introductory half-day workshop
Facilitated by: Ian J Seath

 Apply some basic principles
for displaying tables of data
 Select appropriate types of
chart to analyse and present
data
 Decide how big a sample to
choose in order to be
confident in the results
 Explain why an “average”
could be very misleading

MOST PEOPLE HATE MATHS!
“A Mathematician is a device for turning coffee into theorems.”
Paul Erdos (Hungarian Mathematician)

 An aeroplane flies round the four sides of a 100
mile square
 It flies at 100 mph on side 1, 200 mph on side
2, 300 mph on side 3 and 400 mph on side 4.
 What is its average speed?
100 m.p.h.
300 m.p.h.
200 m.p.h.400 m.p.h.
100
miles
square

The Golden Rules of Measurement
 No measurement without recording
 No recording without analysis
 No analysis without action

PRESENTING TABULAR DATA
“As you can clearly see…”

Badly presented data makes it hard to
understand & improve performance
 Very few people need absolutely accurate numbers (Actuaries,
Accountants, Scientists and Engineers are common exceptions)
 So, for most management information, rounded data will be easier to
handle
 Which of these is easier to identify the biggest percentage increase
in orders?
Orders for product ABC have increased from 4,725 to 6,321 p.a.
whereas DEF orders have increased from 3,015 to 4,643 p.a.
ABC orders have increased from 4,700 to 6,300 whereas DEF orders
have increased from 3,000 to 4,600 p.a.

Tip 1: Round to “2 effective digits”
 It’s much easier on the eye and to do a bit of
mental arithmetic on the second example and
say that an increase of 1,600 ABC Orders is
about a third (33%) and an increase of 1,600
DEF Orders is about half (50%)
 Applied with common sense, rounding to two
effective digits usually makes numbers
easier to cope with and to get a quicker
understanding of what’s going on

Which is easier to read?
Sales (£k) Profit (£k)
2014 25,000 2,400
2013 22,000 2,000
2012 18,000 1,600
2011 14,000 1,000
2010 10,000 650
2010 2011 2012 2013 2014
Sales (£k) 10,000 14,000 18,000 22,000 25,000
Profit (£k) 650 1,000 1,600 2,000 2,400

Tip 2: Columns of data are almost
always easier to read than rows
 Put the latest data, or the biggest numbers, at
the top of the table
 You may not be able to do this with time-based
data
 Columns of data allow the eye to scan up and
down more easily

How would you improve this?
Sales (£k) Q1 Q2 Q3 Q4
Customer A 34.4 32.1 27.7 32.2
Customer B 148.6 139.6 144.3 166.5
Customer C 305.7 284.4 245.3 377.8
Customer D 25.8 29.2 24.9 27.8
Customer E 256.7 242.1 212.9 243.0
Customer F 68.5 73.3 67.9 84.6

Better?
Sales (£k) Q1 Q2 Q3 Q4 Average
Customer C 310 280 250 380 310
Customer E 260 240 210 240 240
Customer B 150 140 140 170 150
Customer F 69 73 68 85 74
Customer A 34 32 28 32 32
Customer D 26 29 25 28 27
Average 142 132 120 156 140

What can you conclude from this?
# of Orders Q1 Q2 Q3 Q4
Customer A 370 350 320 350
Customer B 160 150 150 180
Customer C 47 51 46 63
Customer D 42 40 36 40
What else would you want to know?

Better?
# of Orders Q1 Q2 Q3 Q4 Average
Customer A 370 350 320 350 348
Customer B 160 150 150 180 160
Customer C 47 51 46 63 52
Customer D 42 40 36 40 40
Average 155 148 138 158 150

Tip 3: When to use Tables for data
 Use tables when you have small data sets or
if you need people to see the exact numerical
values in your results
 Round the data to two effective digits unless
readers need the precise numbers

6 basic rules from Prof. ASC Ehrenberg
 Andrew Ehrenberg was a statistician and marketing scientist
 For over half a century, he made contributions to the methodology of
data collection, analysis and presentation
 Ref: Rudiments of Numeracy
 http://www.maths.leeds.ac.uk/~sta6ajb/math1910/p4.pdf
1. Rounding to 2 effective digits
2. Row and column averages
3. Figures are easier to compare in columns
4. Order rows and columns by size
5. Spacing and layout
6. Graphs vs. Tables

HOW MUCH DATA DO YOU
NEED?
“Anecdotes are not statistics.”

Sampling
 In many cases, we obtain data through
sampling; often because it is simply not possible
to measure every single item, or to log every
activity, transaction or incident
 The purpose of sampling is to collect an
unbiased subset which will give you a
manageable amount of data
 When you take samples, they should be
representative (statistically valid and reliable)
and economic to collect (quick and cost-
effective)

Population vs. Sample
Customer Satisfaction
Unhappy Happy
If we surveyed every single
customer over a year to find
out how happy they were with
our services, this is what we
might find.

One person’s sample of 10 customers
Unhappy Happy
How happy are
customers according
to this sample?

Another person’s sample of 10 customers
Unhappy Happy
How happy are
customers according
to this sample?

The “right answer” depends on sample size
Unhappy Happy
Your ability to be confident
about Customer satisfaction
depends on sample size.
If you pick too small a sample
you could, purely by chance,
find very different results and
draw the wrong conclusions.

http://www.surveysystem.com/sscalc.htm

Terms you need to understand
 Confidence Interval (Margin of Error)
 The plus-or-minus figure usually reported in
newspaper or television opinion poll results
 If you pick a CI of 5 and 83% of your sample picks
‘Happy’, you can be “sure” that the 78-88% of the
entire population would have picked ‘Happy’
 Confidence Level
 Tells you how “sure” you can be that the population
would pick an answer within the Confidence Interval
 A 95% CL is most commonly used and means, for the
example above, you can be 95% sure that the true
population is between 78 and 88%

Example
+ or - 3
6000 orders p.a.
= 75 customers per month
You might, therefore, say if 83% of
Customers are ‘Happy’:
“We are 95% confident that between
80 and 86% of customers are Happy”
You can also work out
the CI for a known
sample size

Tip 4: Sampling guidelines
 With static populations (e.g. customers, staff), use
random sampling; for example using Random Number
Tables to decide what (and when) to sample
 Random sampling means that every unit in a population will have
an equal probability of being chosen in the sample
 With time-based data, collect data in sub-groups of 5
values, equally spaced in time (e.g. services are
delivered, or transactions are carried out continuously
over a period of time – call handling in a contact centre)
 If it is not feasible to take sub-groups, take individual values at
regular intervals; e.g. every 10th or 100th

To make meaningful comparisons…
 We must all be measuring the same thing
 Definition of the Performance Indicator
 e.g. an average customer satisfaction score of 8.5
(out of 10) is not the same as 85% of customers
are “satisfied”
 We must be collecting statistically valid
samples

THE “MISLEADING” AVERAGE
“Lies, damned lies and statistics.”

On average
our rope is 2
cm thick !
That’s good
to know !
Supplier
Customer

Do you know what “average” means?
 The length of time (in days) taken for 10
customer orders to be despatched was
recorded
 What was the average time it took (from
order placed to despatch)?
Order
1
Order
2
Order
3
Order
4
Order
5
Order
6
Order
7
Order
8
Order
9
Order
10
6 6.5 7 7 7 7.5 8 8 10 13

Mean, Median and Mode
 Arithmetic Mean - the sum of values divided by the number of
values, often called “the average” (8.0 in our example)
 Median - the middle value when all the values are arranged in order
[or the mean of the two middle values if there is an even number in
the list] (7.25 in our example)
 Mode - the most frequently occurring value (7 in our example)
If the Mean = the Median, the data is distributed symmetrically
The Median and Mode are not affected by extreme values in a set of
data, unlike the Mean

Which “average” would you use & why?
16 data points23 data points
Here are two “response time” histograms

You also need to understand Variation
Bell-shaped Skewed
PlateauBi-modal

What a Histogram might tell you
 Bell-shaped - a symmetrically shaped distribution which typically
represents data randomly distributed, but clustered around a central
value
 Positive or negative skews - where the average value of the whole
set of data is to the left (-) or right (+) of the central value. Look out
for specification limits at the boundaries of the distribution which
might be causing data to be dropped from the population. More
extreme shapes are also known as “precipices”
 Bimodal - where there are two peaks. Usually indicates two sets of
data (e.g. two teams or locations), with different Means have been
mixed
 Plateau - occurs where several sets of data have been mixed (e.g.
from a number of customers/locations/groups)

GRAPHS AND CHARTS
“A picture paints a thousand words.”

Graphs and Charts

Tip 5: When to use Graphs for data
 Use graphs when you have more than ten
data points, or if you want to show people
“the big picture”, not detailed data
 Use graphs when you need to show trends,
over time
 Don’t clutter a graph with too many different
sets of data; it’s usually better to split the data
into separate graphs

Pie Charts
 The data points in a Pie
Chart are displayed as a
percentage of the whole
pie
 Good for: showing
proportions, at a glance
 No good for: showing
trends or comparisons
over time

Bar Charts
 In Bar Charts, categories are
typically organised along the
horizontal axis and values up
the vertical axis
 Bar Charts illustrate
comparisons among individual
items, but do not show
proportions as in a Pie Chart
 Good for: showing quantities of
responses in different
categories; often best when
sorted into biggest to smallest
 Not good for: showing data over
time (use a Line Graph instead)

Histograms
 In Histograms, a variable (e.g.
Time, Length, Height) is
displayed along the horizontal
axis and frequency up the
vertical axis
 Good for: showing the variation
in a set of data and to help
decide if the Mean or Median
are the best choice of average
to quote
 Not good for: showing variations
over time
 N.B. Excel also calls these “Bar
Charts”

PARETO ANALYSIS
“Separate the vital few from the trivial many.”

Pareto Analysis
20%
80%
 80% of problems or errors are often due to only
20% of the causes (The “Vital Few”)
 The remaining 80% of causes account for only
20% of the problems or errors (The “Trivial
Many”)
CausesProblem
Occurrences
Also known as the 80:20 rule
20%
80% The “Vital
Few”
Causes
The “Trivial
Many”
Causes
Most of
the
problems

Pareto Diagram
 A Pareto Diagram is a particular
type of Bar Chart
 Category data is presented in
decreasing size, from left to
right and a Cumulative % line is
also drawn
 Good for: showing the 80:20
Rule – highlighting the few
categories that account for the
majority of performance or
issues
 Not good for: showing data over
time (use a Line Graph instead)

Example: Reasons for delayed orders
Causes Frequency Cumulative
Frequency
% Cumulative
%
A. Unclear supporting
information
47 47 54 54
B. Staff absenteeism 18 65 21 75
C. Non-approved
signatory (> £500)
6 71 7 82
D. Customer not
informed of date
6 77 7 89
E. Computer problems 5 82 6 95
F. Quotations policy 3 85 3 98
G. Other 2 87 2 100

Example: Reasons for delayed orders

IS PERFORMANCE IMPROVING?
“Two data points do not indicate a trend.”

Line Graphs
 In a Line Graph, time data is
distributed evenly along the
horizontal axis, and all value
data is distributed up the vertical
axis
 Good for: showing how results
have changed over time (trends)
 Not good for: comparing lots of
different sets of results (too
many lines make it hard to see
what's going on)
 N.B. Excel enables you to
overlay a statistically derived
trend line

What can you conclude from this Line Graph?
Are weekly Orders increasing, decreasing, or not changing?
No.ofOrders
Mean

4 Week Moving Average
Weekly orders are decreasing (from around week 10)
No.ofOrders

4 Week Moving Average
No.ofOrders

Control Charts
 Statistical Process
Control Charts are a
particular type of Line
Graph
 They enable you to
determine whether
variations are due to
“Special” or “Common”
causes
 The Control Limits are
based on process
variation (s), not
specification tolerances

WORKSHOP REVIEW
“Learning points for action at work?”

ian.seath@improvement-skills.co.uk
07850 728506
@ianjseath
uk.linkedin.com/in/ianjseath
Prepared by
Ian J Seath
Improvement Skills Consulting Ltd.

Making sense of numbers - a half-day workshop

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