Transcript Management Math for Libraries An Infopeople Workshop Jeanne Goodrich [email protected] Winter 2004-05 AGENDA • Numbers Issues • Percentages and Trends • Descriptive Statistics • Making Management Decisions • Your.

Management Math for Libraries
An Infopeople Workshop
Jeanne Goodrich
[email protected]
Winter 2004-05
AGENDA
• Numbers Issues
• Percentages and Trends
• Descriptive Statistics
• Making Management Decisions
• Your Action Plan
What Is It?
Math anxiety is a
feeling of intense
frustration or
helplessness
about one’s ability
to do math.
Math Anxiety
• Primarily an American phenomenon
• But, not a new one
• Real or a myth?
• Parents pass the phobia on to their kids
• Math phobia takes hold gradually
Math Proficiency Declines
• 32% of fourth graders are proficient
• 29% of eighth graders
Percent Proficiency
• 17% of 12th graders
35
30
25
20
15
10
5
0
4th grade
8th grade
12th grade
We’ve Been Set Up!
Until pulled from the
market, Teen Barbie
was programmed to
say “Math is hard!”
Those who supervise,
plan, and manage in
fields such as social
work, librarianship,
retail sales, school
administration, and even
publishing now need to
be familiar with or at
least willing to learn more
math.
Dealing With Math Anxiety
• Recognize as emotional response (often
learned)
– unconstructive ways of addressing
• rationalization
• suppression
• denial
– constructive way to address
• identify sources
• accept the feeling without self-criticism
• learn strategies
Strategies for Approaching Math
Problems
• Define the problem, the number and
operations to be used
• Use smaller numbers
• Examine related but easier problems or
related but more general problems
• Perform calculations in head, with paper,
pencil or calculator
• Work backwards from the solution
• Draw pictures and diagrams
Issues in Library Math
• How are we doing?
• How do we compare to others?
– How much more or less are we doing?
• How much does it cost to do what we do?
– How much could we save if we didn’t
do it?
– How much do alternatives cost?
• How do I interpret this report?
• What else??
Percentages
• Percentages show
– parts of a whole (1/4 or 25%)
– part of a group (1 in 4)
– a ratio (1:4)
– result of division (4 divided into 1)
1

4
Fractions and Percents
You can convert between fractions and
percents:
3 = .15 or 15% Divide 3 by 20 on your
20
calculator
36% = 36 = 9 x 4 = 9
100 25 x 4 25
Reduce to
lowest terms
Translating Percent Problems
Percent problems are like story
problems…you have to translate the
problem statement into a math problem.
30% of what number is 16?
Translation: 30% of •
= 16
This involves solving for the unknown.
Uh, Oh…algebra!
Algebra is basically
an approach to
problem solving
which frequently uses
letters to represent
numbers and
performs arithmetic
operations on them.
Solving for X
• Equation – a mathematical statement that
tells us that one side is equal to the other:
3+4=7
• If you do something to one side, you have
to do it to the other, to keep them equal:
3+4+3=7+3
• Holds true for subtraction, multiplication and
division, also.
3+4=7
5
5
How it Works
If x - 6 = 2, how much is x?
To isolate 6, add 6 to both sides of the
equation: x – 6 + 6 = 2 + 6
x=8
Check it out: 8 – 6 = 2

If 2x = 50, how much is x?
2x = 50
2
2
x = 25
If x = 9, how much is x?
2
x*2=9*2
x = 18
If 2x + 5 = 10, how much is x?
2x = 5
x=5
2
x = 2.5
An Application…
Branch circulation at the Tree County Library
totaled 5,397,892 last fiscal year. The
circulation at the Main Library was one third of
the total circulation. What was the total
circulation for the system?
5,397,892 = 2x
3
5,397,892 x 3 = 2x
16,193,676 = 2x
x = 8,096,838
Some Typical Questions
• What is 25 percent of 24?
• 6 is what percent of 24?
• 6 is 25 percent of what?
Displaying Percents Graphically
Pie Charts
• Show segments
of a whole
• Present only one
variable
Trends
• Trends show where we’re going and where
we’ve been
• In our culture, upwards, left to right is
generally good
• Develop a time series (longitudinal)
by tracking statistics
at regular intervals over a
period of time
Circulation Trends
Total Circulation
9000000
8000000
7000000
6000000
5000000
4000000
3000000
2000000
1000000
0
FY
85/86
FY
86/87
FY
87/88
FY
88/89
FY
89/90
FY
90/91
FY
91/92
FY
92/93
FY
93/94
FY
94/95
FY
95/96
Circulation/Expenditures Comparison
25000000
20000000
15000000
Total Circulation
Operating Expenditures
10000000
5000000
0
FY
85/86
FY
86/87
FY
87/88
FY
88/89
FY
89/90
FY
90/91
FY
91/92
FY
92/93
FY
93/94
FY
94/95
FY
95/96
Circulation/Expense Comparison
9,000,000
25000000
8,000,000
20000000
7,000,000
6,000,000
15000000
5,000,000
4,000,000
10000000
3,000,000
2,000,000
5000000
1,000,000
0
0
FY FY FY FY FY FY FY FY FY FY FY
85/86 86/87 87/88 88/89 89/90 90/91 91/92 92/93 93/94 94/95 95/96
Total Circulation
Operating Expenditures
Trends and Forecasting
• Don’t just put a trend line on your historical
data and run it into infinity
• Consider internal and external influences
• budget
• politics
• changes in population
• new facilities
• changes in services, hours, collection
• user habits and expectations
Descriptive Measures for
Interpretation and Analysis
• Distributions
– averages
– spread
– shape
• Measure of variability
– percentages
– standard deviation
Averages – Measures of Central
Tendency
• Mean – arithmetic average, sum of all
numbers divided by the number of
numbers in the series
• Median – number in the middle of a
data series
• Mode – the number occurring most
often
Picking an Average to Use
• Mean – sensitive to
extreme values
• Median – more
representative of a
range of values,
particularly if great
variance
• Mode – describes
relative popularity of
specific values; only
appropriate measure
for categorical values
Spread or Dispersion
• Range – gives you a picture of the top and bottom
of your data, lets you see how it might be skewed
– Average becomes more meaningful with this
information
• you may want to lop off the outliers (extremely high or
low values)
– Can divide up into quartiles, deciles, or
percentiles (quarters, tens and hundreds)
Quartiles Example
2,500,000
Bottom
25%
Top
75%
1,875,000
1,250,000
625,000
L C L TJ H S L T O P B D L M ID D
O
C RO NW S GR HL SE HG NP CA AL WO BE GS M HW
/
IR
A
F
Using Percentiles
Measurement
Population of
service area
Total operating
expenditures
Operating
expenditures per
capita
Circulation per
capita
Total Circulation
Percentile for
Actual Value
Tree County PL
85th percentile
315,418
70th percentile
$4,303,242
15th percentile
$ 13.64
16th percentile
3.47
71st percentile
1,094,744
2d Approach to Spread
Standard Deviation – a better measure
– Deviation is the distance between any
measurement in a set of data and the
mean of the set. Standard deviation
summarizes the degree to which the
numbers in the distribution differ from the
mean.
– Rule of thumb: two thirds of a distribution
is enclosed within one standard deviation,
95% within two, nearly everything within 3
Histogram
Histogram
9
Positive Skew,
skewed to the
right
8
6
5
4
3
2
1
M
or
e
15
59
96
1.
75
11
43
95
3.
5
72
79
45
.2
5
0
31
19
37
Frequency
7
Determining the Z Score
You can compute how many standard
deviations above or below the mean a
value is:
z = (x - ) / 
x = specific element in the data set
 = the mean
 = the standard deviation
Why Does This Matter?
• Averages define a point around which
other values tend to cluster
• Measures of variability indicate how
widely the values are dispersed around the
average
– Range shows difference between
highest and lowest
– Variance measures amount of dispersion
in a distribution
– Standard deviation measures
dispersion of scores around the mean
Correlation
• Measures the strength of the relationship
between two sets of data (two variables)
• Will always be between –1 and +1
• Positive = when value of one increases, so
does the other
• Negative = when value of one increases the
other decreases
• The nearer r is to zero, the weaker the
correlation
Correlation Example
Population/Operating Budget
$20,000,000
$18,000,000
$16,000,000
Operating Budget
$14,000,000
$12,000,000
$10,000,000
$8,000,000
$6,000,000
$4,000,000
$2,000,000
$0
0
50,000
100,000
150,000
Population
200,000
250,000
300,000
Turnover Rate/Coll Size
25. 0
Turnover Rate
20. 0
15. 0
10. 0
5. 0
0. 0
0
20, 000
40, 000
60, 000
80, 000
100, 000
120, 000
Collection Size
140, 000
160, 000
180, 000
Scatter Plots
Important Warning!
• Correlation is NOT
the same as cause
and effect
• Cause and effect is
tricky stuff:
– A causes B
– B causes A
– C causes A and
B
– Chance
Data Gathering and Analysis
Why do it?
• Measure and improve library services
• Shape policy
• Set goals and performance objectives
• Formulate strategic goals and objectives
• Justify existing or new services
Sampling
• Sampling is the process whereby items are
selected from a population
• The items are then analyzed in order to
generalize the findings to the population
• Samples should be representative of the
population
• Need to collect enough to accurately reflect
full population
Why Sample?
• Less time consuming
• Less costly
• Some populations very large
• Some inaccessible
• Trial periods, pilots are forms of samples
Possibilities
• Work processes
– check-ins per hour
– items processed per hour
– items shelved per hour
• Equipment
– uptime of public pcs
– temperature throughout the building
• Customer attitudes, satisfaction, use of services
– hours of service
Sample Design
• Nonprobability
neither known nor
equal to probability
– convenience
– haphazard
– judgment
– purposive
– quota
– systemic
• Easier, less costly
• Probability
each element in
population has equal
chance of inclusion
– random
– stratified
• Takes more time, more
accurate
Sample Size
Complicated formula boils down to:
– 10% of population unless very large
– No more than 1000 needed, regardless
of population size
– Very small samples likely to have more
errors and be less representative
Key Concepts
• Confidence interval – range around which
actual value for the population is likely to
fall. Stated +/- 5, for example
43% of those surveyed indicated they
would support the measure = 38-48% of
the population
• Confidence level – Tells you how sure you
can be, how often the true percentage of
the population would pick an answer within
the confidence interval. 95% usually used.
• Statistical significance – probably true,
small probability of happening just by
chance
Summarizing Categorical Data
• Categorical data captures qualities or
characteristics (gender, age, race, ethnicity,
language spoken at home)
• Often summarized by reporting
percentages in various categories
• Cross Tabs or Two-way tables summarize
information from two categorical variables
at once, such as age and language spoken
at home
Regression Analysis
A way of defining the extent to which two
variables are related. Regression can be
used in an attempt to predict things—but it
can be tricky.
For example, you may have a community
survey and find, through regression
analysis (that your professional surveyor
does for you) that your Internet policies on
filtering are a convincing reason to vote
“No” to 25% of respondents
Make Your Data Understandable
• You’re not alone in not liking numbers
• Dense text obfuscates…as do unfamiliar
words
• Think about your audience when choosing
how to present your data
• Use charts and graphics
• Be sure you make the point you intend to
make
• Aggregate and disaggregate for your reader
This LOOKS like
more, but is really
less
OCLC report: Libraries: How
They Stack Up
www.OCLC.org/index/compare/
Aggregation & Disaggregation
• “Roll up” data to make it more
understandable
– individual branch, department or unit
statistics into a summary
– monthly into yearly
– five and ten year summaries of key
statistics
• “Drill down” to get more detail
– identify variables
– identify data errors or collection errors
library material
Video
Multnomah
County Library,
FY 04
Compiled from
holdings and
circulation
reports; turnover
computed.
Clear information
from several
reports.
circulation holdings turnover
(dynix only) (excl. Lost)
rate
1,934,571
75,639
25.6
DVD
816,935
35,731
22.9
CD-ROM
109,696
4,854
22.6
2,052,166
114,326
18.0
Audio
Board books &
catalogued Pb
748,298
45,663
16.4
334,186
21,116
15.8
Juvenile Fiction
651,668
61,837
10.5
Picture books (JE)
1,615,009
154,223
10.5
Fiction
2,098,367
207,922
10.1
464,088
46,541
10.0
Non-fiction
6,018,557
673,112
8.9
Juvenile Non-fiction
1,046,573
142,770
7.3
Large Print
165,993
24,653
6.7
Int'l Languages
236,376
39,363
6.0
Music Scores
160,538
44,281
3.6
70,026
247,646
0.3
Total 18,523,047
1,939,677
9.5
CD
Young Adult
Other
Correlation of FTEs and Circ
C o r r el at i o n
of FTE to
C i r i s . 52 2
800,000
700,000
Circulation
600,000
500,000
400,000
300,000
200,000
100,000
0.000
10.000 20.000 30.000 40.000 50.000
FTEs
Financial Analysis
What financial problems have you dealt
with or would you like to be better
equipped to deal with?
Are there aspects of your budget
process that you’d like to understand
better?
Ratios, Rates and Percentages
• Ratio is a fraction that divides two
quantities: 3 girls to 2 boys; 1 manager to
12 staff members
• A rate is a ratio that reflects some quantity
per a certain unit: circulation is 4.5 per
capita, turnover rate is 9.8 per Adult Fiction
title
• A percentage reflect a proportion of the
whole: 15% of the materials budget goes
for large type materials
Cost Analysis
• Cost Allocation – How would you approach
allocating overhead costs to library
programs?
• Cost per Unit – Can you relate outputs to
inputs? Do you know what it costs you to
catalog a book? A set of books on tape?
What variables are important?
Cost Benefit Analysis
A decision-making technique to develop
quantitative information on how to allocate
resources.
cost-benefit ratio: estimated benefits
costs
Objective: benefits outweigh costs
Break Even Analysis
The point where total costs equal total
revenues
Total fixed costs
= Break even point
Contribution margin
If a library buys 300 book bags from a
supplier for $12 each and plans to sell them
for $15, how many do they have to sell to
break even?
Return on Investment
Business concept you might be able to apply
in the library
Rate of return = revenue cost center profit
investment
Library gift shop:
$35,000 = 30%
$117,500
Your Action Plan
• What project will you take on using what
you’ve learned today?
• Have you found some ways to answer your
questions with numbers?
• Have you found some new ways to use
numbers that you hadn’t known of before
today?
• What are you going to do as you prepare
your next budget?
Thank You!
[email protected]