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

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 81.5 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

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

Min, max and range are a start 23 maximum minimum 27 4

Score mean 72 - 81.5 = - 9.5 units from mean Standard Deviation measures how spread out the all the values in the data set are from the mean

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%

[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?

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. ]

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

TEST SCORE EXAMPLE: RESULTS Class A Class B Class quiz Average Standard Deviation 81.5 81.5 4.88 15.91 highly varied scores, wide distribution many similar scores, close to mean

CALCULATING the SD: an example 2. Data (x) (datum-mean)2 2.3 (2.3-3.4)2 = 1.21 3.7 (3.7-3.4)2 = 0.09 4.1Σ (4.1-3.4)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 3. 1.79/(n-1) = 1.79/(3-1) 0.90 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

YOU TRY

Alternative approach: spreadsheet functions B 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)

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!

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

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