Descriptive Statistics II:By the end of this class you should be able to:

• describe the meaning of and calculate the mean and standard deviation of a sample

• estimate normal proportions based on mean and standard deviation

• plot a histograms with alternative scaling

Palm: Section 7.1, 7.2

please download cordbreak1.mat & FWtemperature.txt

Exercise

• Download FWTemperature.txt• Read into MATLAB• Prepare a single figure with two plots

– a histogram of March highs (row 2)– a histogram of April highs (row 4)

• Label these plots fully • Print out the your commands and the

resulting figure

Review: Quantifying Variation

1

)(1

2

n

xxs

n

ii

x

n

xx

n

ii

1

Mean

Central Tendency

>> mean(x)

Standard Deviation

Spread

>> std(x)difference deviation of each point

about the mean

squared all values positive

Summation yields one number

Divide by n-1 normalize the sum

for based on degrees of freedom

Formula MATLAB EXCEL

Mean >> mean(variable) = average(range)

Sample Standard Deviation

>> std(variable) = stdev(range)

n

xx

n

ii

1

1

)(1

2

n

xxs

n

ii

x

-4 -3 -2 -1 0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

standard deviations from the mean

prob

abili

ty d

ensi

ty (s

cale

d fre

quen

cy)

The Normal (Gaussian) Distribution

(Population)

StandardDeviation

Mean

Mode

Note on Sample and Population Statistics

Sample (The estimate from a sample of the whole population)

Population(The true value from the entire population)

Standard Deviation

s

Meanor m

x

s

n

as

x

Expected Proportions for known

-4 -3 -2 -1 0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

standard deviations from the mean

prob

abili

ty d

ensi

ty (s

cale

d fre

quen

cy)

68 %

95.5 %

99.7%

Percentage of observations in the given range

1

2

3

mean

-4 -3 -2 -1 0 1 2 3 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

standard deviations from the mean

prob

abili

ty d

ensi

ty (s

cale

d fre

quen

cy)

68 %

Expected Proportions for known

16 %

%162

68100

Proportions Problem

Data analysis of the breaking strength of a certain fabric shows that it is normally distributed with a mean of 200 lb and a variance (2) of 9.

• Estimate the percentage of fabric samples that will have a breaking strength between 197 lb and 203 lb.

• Estimate the percentage of fabric samples that will have a breaking strength no less than 194 lb.

145 165 185 205 225 245 265 285 305 325 345 3650

1

2

3

4

5

6

7

8

9x 10

-3

Breaking Force (n)

Sca

led

Fre

quen

cyCord Breaking Distribution with Normal Curve

)2/()( 22

2

1)(

xexp

Review: Types of Histograms

Type Freq. Formula Use MatlabAbsolute Frequency

absolute count in each bin

= zfor a quick picture >> hist(x, n)

Relative Frequency

fraction of total count in each bin

compare samples when total counts differ

>> [x,z] = hist(x)>> zr = z/sum(z)>> bar(x, zr)

Scaled Frequency

fraction of total area in each bin

compare samples when bin sizes differs

>> b = bin centers>> [x,z] = hist(x,b)>> zs = z/(sum(z)*w)>> bar(x, zs)

)(zsumz

widthbinzsumz

*)(

Additional Example (not covered in class)

Looking at two sets of data • Look at a histogram of the second set of data,

‘cord2’

• How would you compare it to cord the first set of data?

• What problems do you run into?