STUDENTS WILL DEMONSTRATE UNDERSTANDING OF THE

CALCULATION OF STANDARD DEVIATION AND CONSTRUCTION

OF A BELL CURVE

Standard Deviation&

The Bell Curve

Standard Deviation

1st find the variance for a set of dataVariance is the average squared

deviation from the mean of a set of data

Computing the Variance

Grades from recent quiz in AP Biology:96, 96, 93, 90, 88, 86, 86, 84, 80, 70

1st Step: find the mean:

96 + 96 + 92 + 90 + 88 + 86 + 86 + 84 + 80 + 70 ÷ 10

868 ÷ 10 = 86.8 87

Computing the Variance 2nd Step:determine the deviation from the mean

for each grade then square it: (96 – 87)² = (9)² = 81 (96 -87)² = (9)² = 81 (92 – 87)² = (5)² = 25 (90 – 87)² = (3)² = 9 (88 – 87)² = (1)² = 1 (86 – 87)² = (-1)² = 1 (86 – 87)² = (-1)² = 1 (84 – 87)² = (-3)² = 9 (80 – 87)² = (-7)² = 49 (70 – 87)² = (-17)² = 289

Computing the Variance

now add these 10 #’s and ÷ 10 81 + 81 + 25 + 9 + 1 + 1 + 1 + 9 + 49+ 289 ÷ 10

546 ÷ 10 = 54.6 = variance

Formula for Variance

Standard Deviation

shows the variation in dataif data pts are close together the

standard deviation will be smallif data pts are spread out the standard

deviation will be larger

σ or S = symbol used for standard deviation

The Bell Curve

represents a normal distribution of data & is used to show what standard deviation represents

The Bell Curve

1 standard deviation from the mean (μ is symbol for 1 standard deviation) in either direction on horizontal axis represents 68% of the data

The Bell Curve

2 μ = 2 standard deviations from the mean and will include ~95% of your data

3μ = 3 standard deviations form the mean and will include ~99% of your data

Standard Deviation Calculation

Standard Deviation is the square root of the variance

Back to the Example Quiz Grades

Mean = 87Variance = 55

σ = √55 = 7.42 = 7.4

Mean = 87 so (87 -7.4) & (87 + 7.4) is the range of 1 standard deviation

Example Continued

My range for 1 standard deviation is : 79.6 to 94.4 80 to 94

in my class scores 93, 90, 88, 86, 86, 84, 80 are w/in 1 standard deviation of the mean

check: 1 standard deviation is where 68% of data should fall: my class scores: 70%

If data are a sample of larger group:

Use the formula:

for my class data:

S = √variance ÷ n – 1S = √55 ÷ 9 = √6.1 =2.5 which means the

measurements vary by +/- 2.5 from the mean

so for my class data:Mean: 87S = 2.5

1 µ would be (87 – 2.5) thru (87 + 2.5) or 84.5 to 89.5

2 µ would be (87 – 5) thru (87 + 5) 3 µ would be (87 – 7.5) thru (87 – 7.5)

How Can I Use S?

Suppose instead of grades my data described the average height of a certain plant (in cm). We can use S to predict the probability of finding a plant at a particular height:

What is the probability of finding a plant that is 100 cm tall?

100 cm falls in the 3rd standard deviation range

from a bell curve we know that represents ~2% of a population so… There is a 2% chance a plant will grow to 100cm.

Make Friends with the Bell Curve