Statistics and Spreadsheets Harris Chapter 4

Gaussian DistributionConfidence Intervals

Student’s T-TestsQ-test

Control ChartsSpreadsheets

Gaussian Distribution (random!)

• Mean Value:– The arithmetic

“average”– For a set of data, the

closer your mean is to the true value, the more accurate your results are!

n

X X

i

0i∑

=

Standard Deviation (reproducibility)

• Standard deviation is based on the fact that you will assume that errors are the result of RANDOM events.

• It is based on the shape and distribution of the Gaussian Curve

• A smaller standard deviation means that your results are more reproducible (they don’t vary as much from measurement to measurement).

The Gaussian Curve• Plotting of random

events• Defines standard

deviation• Has a mathematical

definition (formula for the curve)

• Discussed in more detail in the text

99.9 %+/- 3 STD DEV

95.5 %+/- 2 STD DEV

68.3 %+/- 1 STD DEV

% of Events Affected by Random Error that Occur

# of Standard Deviations from the Mean

Calculating a STD DEV (by hand)• Based on the difference

between each value and the mean.

• Also based on the degrees of freedom– Number of measurements

minus one– n-1

1-n

)x(x s

i

0

2i∑ −

=

Let’s do it manually once, together!

• M&M’s Results Handed Out to All!• Calculate mean and standard deviation• Setup a simple table • Use table to keep track of the squared

terms!• LEARN TO DO THIS USING YOUR

CALCULATOR AND MSEXCEL (STDEV is the correct function)

Confidence IntervalsHow Certain Are You?????

• Confidence intervals allow us to calculate a range of values in which we can be confident, at some level, that the “true” value lies

• Originally based on the growth of yeast in beer!• One of the most important tools in evaluating

data!• Back to Elementary School: draw a number line to

see how this works!

Calculating a Confidence Interval

• Determine the Mean • Determine the Standard

Deviation• Determine the degrees of

freedom (n-1)• Decide how confident

you want to be in your data (80%, 90%, 95%, etc.)

• Calculate using appropriate formula.

ns t x ×

±=µt is the value of Student’s t from a t-table (Figure 4-20

n is the # of observations

s is the standard deviation

Confidence Interval Calculation: John C. SchaumloffelCalculate the [Zn] at the 95% confidence interval[Zn] ppm

1.20 u = mean +/- (t x s)/(n^0.5)1.401.501.101.101.26 mean

0.1817 STDEV5 n4 n-1 (degrees of freedom)

2.776 t-value, n=5, 95% confidentHarris Table 4-2

0.2255 is the range of the confidence interval (the +/- value)

Confidence Interval = 1.26 +/- 0.23 ppm Zn

Therefore, we are 95% confident that the "true" value for the concentrationconcentration of Zinc is between 1.03 and 1.49 ppm.

Comparison of Mean’s w/Student’s T

• We can compare two sets of data to determine how confident we are that they are either– Statistically similar– Statistically different

• This is ONLY a statistical test, you can also rely on– Your intuition as a chemist– Your practical experience

• But, statistical test are what win in court!

• We will concentrate on Harris’ “Case Two”– A quantity is measured multiple times by two different

techniques. Each technique gives a mean and standard deviation for the quantity. Are these similar?

• Steps….– Calculate a pooled standard deviation– Calculate a t-value using the pooled standard deviation– Compare the tcalculated to the correct t-value from the table

(ttable)– If tcalc > ttable, the results are statistically different– If tcalc < ttable, the results are statistically similar

Are the [Pu] in the contaminated soil samples from Chemist #1 and Chemist #2 statistically

different?

Q-test to Eliminate Outliers

• Used when you have a set of data with one or more suspect values (“out of whack”)

• A statistical test you can use to provide evidence to eliminate an outlier from the data set

• ONLY a statistical test….

Are any of the soil [Pu] values outliers? Lets check using the Q-

test.

Control Charts

• A graph showing the mean value for a result collected over a period of time

• Ranges for +/- 1, 2, 3 or more standard deviations are shown on the graph

• Used to visually see if data are falling out of a range which would be defined by RANDOM error– Instrumental Fluctuations– Standards or Samples Degrading– Instrument Operator Changing….

• In most regulatory and industrial settings, the mean +/- 2 STDEV is considered acceptable– Warning limit

• Outside of +/- 2 STDEV is considered the action limit– You must correct the situation in this case…..

• Usually, repeated analysis of a known standard is used to develop a control chart.

[Hg] in Quality Control Sample….

Day [Hg] ppb UWL LWL UAL LAL MEAN 0.1431 0.1 0.280937 0.005063 0.349906 -0.06391 STDEV 0.0689692 0.12 0.280937 0.005063 0.349906 -0.063913 0.12 0.280937 0.005063 0.349906 -0.063914 0.13 0.280937 0.005063 0.349906 -0.063915 0.08 0.280937 0.005063 0.349906 -0.063916 0.09 0.280937 0.005063 0.349906 -0.063917 0.11 0.280937 0.005063 0.349906 -0.063918 0.17 0.280937 0.005063 0.349906 -0.063919 0.2 0.280937 0.005063 0.349906 -0.06391

10 0.31 0.280937 0.005063 0.349906 -0.06391

[Hg] Control Chart (spectrophotometry)

-0.1

0

0.1

0.2

0.3

0.4

0 2 4 6 8 10 12

Analysis Day

[Hg]

ng/

mL