Mathematical Terms
Related to a Group of
Numbers
Data Collection
Mean, Mode, Median, & Range
Standard Deviation
Mean = the average
• Consider your measurements as a set of numbers.
• Add together all the numbers in your set of
measurements.
• Divide by the total number of values in the set
Ages of cars in a parking lot:
20 years +10 years +10 years +1 year+15 years +10 years=
66 years/6 cars = 11years
(answer is a counted numbers for sig figs are not an issue)
• Note: the mean can be misleading if there is one very high
value and one very low, the average will be high)
Median
The number in the middle
Put numbers in order from lowest to highest
and find the number that is exactly in the
middle
20, 15, 10, 10, 10, 1
Since there is an even number of values the
median is 10 years (average of the 2
middle values)
Or to determine the Median:
20 years, 15 years, 13 years, 11 years,
7 years, 5 years, 3 years
With an odd number of values the median is
the number in the middle or 11 years
Mode
• Number in data set that occurs most often
20, 15, 10, 10, 10, 1
• Sometimes there will not be a mode
20, 17, 15, 8, 3
Record answer as “none” or “no mode” –
NOT “0”
• Sometimes there will be more than one
mode
20, 15, 15, 10, 10, 10, 1
Range
The difference between the lowest and
highest numbers
20, 15, 10, 10, 1
20-1 = 19 years
The range tells you how spread out the data
points are.
Example:
100% 78% 93% 84% 91% 100% 82% 79% 80%
What is the mean?
What is the median?
What is the mode?
What is the range?
Measured Values
When making a set of repetitive measurements,
the standard deviation (S.D.) can be
determined to
•indicate how much the samples differ
from the mean
•indicate also how spread out the values of
the samples are
Standard Deviation
• The smaller the standard deviation, the
higher the quality of the measuring
instrument and your technique
• Also indicates that the data points are also
fairly close together with a small value for
the range.
• Indicates that you did a good job of
precision w/your measurements.
A high or large standard deviation
• Indicates that the values or measurements
are not similar
• There is a high value for the range
• Indicates a low level of precision (you
didn’t make measurements that were
close to the same)
• The standard deviation will be “0” if all the
values or measurements are the same.
Formula for Standard Deviation
=
(highest value – lowest value)
range
N
=
N
N = number of measured values
As N gets larger or the more samples
(measurements, scores, etc.), the reliability of
this approximation increases
Example 1
22.5 mL, 18.3 mL, 20.0 mL, 10.6 mL
The Standard Deviation would be:
(highest value – lowest value)
range
=
N
=
N
Range = 22.5 – 10.6g = 11.0 mL
N=4
11.0g
4
= 5.95 mL
= 6.0 mL
S.D. = 17.9 ± 6.0 mL (expressed to the same
level of precision as the mean)
Example 2
Age of cars:
15 years, 12 years, 12 years, 1 year, 14 years
and 12 years
What is the mean?
What is the range?
What is N ?
The Standard Deviation would be:
(highest value – lowest value)
range
=
N
=
N
14
 5.726091 years
6
S.D. = 11 ± 6 years.
Expressing the S.D.
S.D. represents the uncertainty of the last
sig fig of the set of data
Therefore the S.D. is expressed to the same
level of precision as the mean value
234 (ones place)
17.9 (tenths place)
~Review~
•
•
•
•
Mean – average
Mode – which one occurs most often
Median – number in the middle
Range – difference between the highest
and lowest values
• Standard Deviation – The standard
deviation of a set of data measures how
“spread out” the data set is. In other
words, it tells you whether all the data
items bunch around close to the mean or
is they are “all over the place.”