# Uncertainty in Measurements How to handle human measuring errors

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```					Uncertainty in Measurements
How to handle human measuring errors
Choosing the Correct Piece of
Equipment
Every measuring
device you use
has a scale
(lines) on it.
Choosing the Correct Piece of
Equipment
The scale
determines how
precise your
measurement
will be.
Choosing the Correct Piece of
Equipment
Your first step is
always figuring
out how much
each line
represents.
Choosing the Correct Piece of
Equipment
What volume
does each line
represent on
cylinder?
Choosing the Correct Piece of
Equipment
1 mL is correct!
Liquids in glass
containers curve
at the edges
forming a…
Liquids in glass
containers curve
at the edges
forming a…
MENISCUS
Bend down to get
eye level with
bottom of the
MENISCUS
Making a Measurement
If a quantity falls
between the lines on
the scale, you must
estimate some of the
measurement
Making a Measurement
Estimating introduces
human error into the
measurement
It is common practice
to report all the
“certain” digits (the
ones you have a line
for)

36 mL
Plus one more decimal
place beyond the
scale

36.3 mL
It is meaningless to
report this level to
the hundredths digit
because you are
the tenths digits.

36.35 mL
What is the Volume Here?

36.5 mL
What is the Volume Here?

42.9 mL
What is the Volume Here?

47.0 mL
Be Careful What You Write
Acceptable uncertainty is plus or minus “1” of
the last digit recorded.

Ex. 1.5 lbs. means the measurement is
between 1.4 and 1.6 lbs.

Ex. 1.500 lbs. means the measurement is
between 1.499 and 1.501 lbs. This conveys
a much more precise measurement!!
Significant Figures
All of the certain digits plus the first uncertain
digit (the estimated number) are referred to
as the Significant Figures of the
measurement.
Rules For Counting Significant
Figures
1.    Nonzero numbers are always significant.

Ex. 32,127     has 5 sig.figs.

Ex. 24         has 2 sig.figs.
Rules For Counting Significant
Figures
1.    Nonzero numbers are always significant.
2.    All numbers in scientific notation are significant.

Ex. 6.02x1023 has 3 sig.figs.

Ex. 1.392x10-5 has 4 sig.figs.
Rules For Counting Significant
Figures
3. Leading zeros (zeros that precede nonzero digits)
are never significant.

Ex. 0.0025 has 2 sig.figs.

Ex. 0.7624 has 4 sig.figs.
Rules For Counting Significant
Figures
4. Captive zeros (zeros between nonzero digits)
are always significant.

Ex. 34,005 has 5 sig.figs.

Ex. 10.00305 has 7 sig.figs.

Ex. 0.002008 has 4 sig.figs.
Rules For Counting Significant
Figures
5. Trailing zeros (zeros at the end of the number) are
sometimes significant.
YES if you see a decimal point
NO if you don’t see a decimal point.

Ex. 2,000. has 4 sig.figs.

Ex. 2,000 has 1 sig.fig.

Ex. 2,000.00 has 6 sig.figs
Rules For Counting Significant
Figures
6. “Counting” numbers have an infinite number of
significant figures

Example 1: There are 20 students in the room.

Example 2: Definitions…1 foot is exactly 12
inches
How many significant figures?
   0.0105
   0.050080
   8.050 X 10-3
   20,200

 Answers: 3, 5, 4,3
Handling Significant Figures in
Mathematical Operations
Uncertainty accumulates as calculations are
carried out so we must follow a set of rules
to handle significant figures when
multiplying, dividing, adding, and
subtracting numbers.

I’m uncertain!!
When Multiplying or Dividing
Your answer can NOT have more sig. figs.
than the least precise measurement

Ex.      4.56       3 sig.figs.
x 1.4        2 sig.figs.
6.384      can only have 2 sig.figs
When Multiplying or Dividing
Your answer can NOT have more sig. figs.
than the least precise measurement

Ex.        4.56          3 sig.figs.
x 1.4           2 sig.figs.
6.384         can only have 2 sig.figs

You are limited to 2 sig.figs. by the measurement
1.4, so you must round 6.384 to 6.4
When Multiplying or Dividing
Your answer can NOT have more sig. figs.
than the least precise measurement

Ex.        4.56          3 sig.figs.
x 1.4           2 sig.figs.
6.4           can only have 2 sig.figs

You are limited to 2 sig.figs. by the measurement
1.4, so you must round 6.384 to 6.4
When Adding or Subtracting
Your answer can’t have more places past the
decimal than the least precise measurement

Ex.       12.11      2 places past decimal
+ 18.0       1 place past decimal
1.013      3 places past decimal
31.123     can only have 1 place
past decimal
When Adding or Subtracting
Your answer can’t have more places past the
decimal than the least precise measurement

Ex.       12.11       2 places past decimal
+ 18.0        1 place past decimal
1.013       3 places past decimal
31.123      can only have 1 place
past decimal
You must round answer to only 1 place past decimal
When Adding or Subtracting
Your answer can’t have more places past the
decimal than the least precise measurement

Ex.       12.11       2 places past decimal
+ 18.0        1 place past decimal
1.013       3 places past decimal
31.1        can only have 1 place
past decimal
You must round answer to only 1 place past decimal
How to Round for Sig. Figs.
Only look at the digit immediately to the right
of the last significant figure

– if less than 5, preceding digit stays the same
– if 5 or more, round preceding digit up by one
You Try It
 Round   4.348 to two sig. figs.

These are the 2 sig figs
You Try It
 Round   4.348 to two sig. figs.

So look at the very next number for rounding
It’s less than 5 so the 3 stays like it is

Final Answer is 4.3

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