1. DESCRIBING A POPULATION
The variation of a trait in a population - or lack therefore – can be described quantitatively using statistics.
mathematical average
the most instances
half of the data points above, half below

2. ANOTHER METRIC : STANDARD DEVIATION
Case Study Two classes took a recent quiz. There were 10 students in each class, and each class had a mean score of Since the means are the same, it is tempting to assume that the students in both classes did pretty much the same on the exam

3. The mean does not tell us anything about the grade distribution of or variation of grades in the population
mean
student scores
Need a way to measure
the spread of grades

4. Min, max and range are a start23
maximum
minimum 27 4

5. Score mean
= units
from mean
Standard Deviation measures how spread out the all the values in the data set are from the mean

6. In a normally distributed population (a bell curve)
+/- 1 SD describes 68% of the population
+/- 2SD describes 95% of the population
+/- 3 SD describes all of the population
anything outside of 3SD is an outlier
68%
95%
100%

7. [SIDEBAR: THE GRADE CURVE]
F D C B A
mean
(aka +3 SD)
Best used with an exam that is difficult and yields a wide range of scores – why?

8. SD Formula
the difference of all of the values from the mean of the population sample, summed, squared, divided by the sample size, and then square-rooted (phew!)
N-1? The standard deviation of a sample of a population uses 1/(N-1), generating an unbiased estimate.
In the rare case when an entire population is counted/measured, the standard deviation calculation uses 1/N. ]

9. Interpreting the SD
If SD is small , the data is close to the mean. Can infer the IV is likely to be affecting the DV
If SD is LARGE, the numbers are spread out from the mean. Other factors are likely influencing the DV
In biology, to describe a population
In education to calculate a grade curve
In production systems as the upper and lower control limit; anything above or below represents a problem.
In finance, the higher the standard deviation, the riskier the investment
For sports teams, a high standard deviation shows that they perform well in some situations but not in others.
Regarding climate while two cities may each have the same average maximum temperature, the standard deviation of the daily maximum temperature for a coastal city will be less than that of an inland city

10. TEST SCORE EXAMPLE: RESULTS
Class A
Class B
Class quiz Average
Standard Deviation
highly varied scores, wide distribution
many similar scores, close to mean

12. CALCULATING the SD: an example2.
Data (x)
(datum-mean)2
2.3
( )2 = 1.21
3.7
( )2 = 0.09
4.1Σ
( )2 = 0.49
Mean ( ) = 3.4
Summed (Σ) = 1.79
Note: If you have multiples of a datum, multiply the (datum-mean)2 by the number of occurrences before summing _ 1. x
/(n-1) = 1.79/(3-1) variance
4. Square root 0.90 = 0.95 standard deviation
5. Mean +/- 1SD = 2.45 – 4.35 min/max that describes 68% of a normally distributed population

13. YOU TRY

14. Alternative approach: spreadsheet functionsB 1
2.3
=(A1-A4)^2 2
3.7
=(A2-A4)^2 3
4.1
=(A3-A4)^2 4
Mean =(SUM(A1:A3)/3)
=(SUM(B1:B3)
Tip: insert the numeric value for the mean in cell B1, then drag and fill the rest of the cells of the column.
variance SD
=(B4/(3-1))
=SQRT(B4)

15. 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 X   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 =
Save in graphing calculator or spreadsheet!

17. S Use Sd to CalCULATE SEM Add SEM to bar graph of mean
standard error of the mean
standard deviation S
SEM
population sample size
mean

18. Use SEM to compare means of data
populations same or different?
More later…