Download Standard Deviation

DESCRIBING A POPULATION
The variation of a trait in a population - or lack therefore
– can be described quantitatively using statistics.
mathematical average
A not quite
‘normal’, or
bell curve data
distribution
half of the data points
above, half below
the most instances
ANOTHER MEASURE : STANDARD DEVIATION
Case in Point:
Two classes took a recent quiz. There were 10
students in each class, and each class had a
mean average score of 81.5
Since the averages are the same, can we
assume that the students in both classes all
did pretty much the same on the exam?
Class A
Mean
Class B
The mean does not tell
us anything about the
grade distribution or
variation of grades in
the population
Need a way to measure
the spread of grades
Mean average
Range (spread)
Mean
and
range
tell only
part of
the
‘story’
Here, most of the numbers hover around the mean, and are not
evenly distributed throughout the range.
mean
72 - 81.5 =
-9.5
Standard Deviation is a measure of how spread out
the values in the data set are from the mean
SD = the difference of all of the values from the mean
of the population, summed, divided by the population
size
SD IS ADDED/SUBTRACTED FROM THE MEAN
Mean +/- the SD defines a range of values.
In a normally distributed population, +/- 1 SD describes
68% of the population
+/- 2SD describes 95% of the population
+/- 3 SD describes 98% of the population
Anything outside of
3SD is an outlier
68%
95%
98%
If SD is small , the data is closer
to the mean. More likely the IV
is affecting the DV.
If SD is LARGE, the numbers
are spread out from the mean.
Other factors are likely
influencing the DV.
[SIDEBAR: THE GRADE CURVE]
F
D
C
B A
mean
Best used with an exam that is difficult
yielding a wide range of scores – why?
RESULTS
Team A
Average on
the Quiz
Standard
Deviation
Using:
Team B
81.5
81.5
4.88
15.91
Variance and Standard Deviation: Step by Step
• Calculate the mean, x.
• Subtract the mean from each observed value
Hint: do each value once, account for multiple later
• Square each of the differences.
multiply by the number of each value now
Sum this column.
• Divide by n -1 where n is the
number of items in the sample. This is the variance.
• To get the standard deviation, take the square root of
the variance.
Example 1.
Data
(datum-mean)2
2.3
(2.3-3.4)2 = 1.21
3.7
(3.7-3.4)2 = 0.09
4.1
(4.1-3.4)2 = 0.49
Mean = 3.4
Summed* = 1.79
2.
1.79/(3-1) = 0.90 variance
3.
Square root of 0.90 = 0.95 standard deviation
4.
Mean +/- 1SD = 2.45 – 4.35 range that describes
68% of a normally distributed population
*If you have multiple incidences of the same
datum, multiply the (datum-mean)2 by the
number of occurrences before summing
Alternative approach: spreadsheet functions
A
B
1
2.3
=(A1-A4)^2
2
3.7
=(A2-A4)^2
3
4.1
=(A3-A4)^2
4
=(SUM(A1:A3)/3)
=(SUM(B1:B3)
variance
=SUM(B4/(3-1)
SD
=SQRT(B4)
Remember you can drag by the bottom left corner to
SUM, or to fill subsequent cells (but you’ll have to
correct the mean cell reference
Alternative approach: graphing calculator
s = square root of [(sum of X2 - ((sum of X) * (sum of X)/N)) / (N-1)]
Step 1: Square each of the scores
X 1 2 3 4 5
X2 1 4 9 16 25
Step 2: Use the x, x2 in formula
= square root of [(55-((15)*(15)/5))/(5-1)]
= square root of [(55-(225/5))/4]
= square root of [(55-45)/4]
= square root of [10/4]
= square root of [2.5]
s = 1.58113
Save in graphing calculator or spreadsheet!