Download Mathematical Terms Related to a Group of Numbers Data Collection

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 (one very high value and
one very low)
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:
The mean of four numbers is 50.5
101
99
What is the median?
What is the mode?
1
1
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
• Indicates 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
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
The results of several masses of an object
weighed by 5 different students on a multibeam balance
11.36g, 11.37g, 11.40g, 11.38g & 11.39g
Average = 56.90g ÷ 5 = 11.38g
SO:
11.36g, 11.37 g, 11.40 g, 11.38 g & 11.39g
The Standard Deviation would be:
(highest value – lowest value)
range
=
N
=
N
Range = 11.40g – 11.36g = 0.04g
N=5
0.04g
5
= 0.0178 g
=
0.02 g
The Standard Deviation from the mean is:
Mean = 11.38 g
SD = 0.02 g
Expressed as 11.38g ± 0.02
This tells us that the measurements were within
two hundredths of the mean – either less than
the mean or greater than the mean.
~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.”
Graphical Analysis
The graphs show three
normal distributions
with the same mean,
but the taller graph is
less “spread out.”
Therefore, the data
represented by the
taller graph has a
smaller standard
deviation.