Module 7: Comparing Datasets and Comparing a Dataset

with a Standard

How different is enough?

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Concepts Independence of each data point Test statistics Central Limit Theorem Standard error of the mean Confidence interval for a mean Significance levels How to apply in Excel

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Independent Measurements

Each measurement must be independent (shake up basket of tickets)

Example of non-independent measurements– Public responses to questions (one result affects

next person’s answer)– Samplers too close together, so air flows

affected

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Test Statistics

Some number calculated based on data In student’s t test, for example, t If t is >= 1.96 and

– population normally distributed,– you’re to right of curve, – where 95% of data is in inner portion,

symmetrically between right and left (t=1.96 on right, -1.96 on left)

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Test statistics correspond to significance levels

“P” stands for percentile Pth percentile is where p of data falls below,

and 1-p fall above

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Two Major Types of Questions Comparing mean against a standard

– Does air quality here meet NAAQS? Comparing two datasets

– Is air quality different in 2006 than 2005?– Better?– Worse?

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Comparing Mean to a Standard

Did air quality meet CARB annual standard of 12 microg/m3?

year Ft Smith avg

Ft Smith Min

Ft Smith Max

N_Fort Smith

‘05 14.78 0.1 37.9 77

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Central Limit Theorem (magic!)

Even if underlying population is not normally distributed

If we repeatedly take datasets These different datasets have means that

cluster around true mean Distribution of these means is normally

distributed!

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Magic Concept #2: Standard Error of the Mean

Represents uncertainty around mean

As sample size N gets bigger, error gets smaller!

The bigger the N, the more tightly you can estimate mean

LIKE standard deviation for a population, but this is for YOUR sample

N

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For a “large” sample (N > 60), or when very close to a normal distribution…

Confidence interval for population mean is:

nsZx

Choice of z determines 90%, 95%, etc.

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For a “Small” SampleReplace Z value with a t value to get…

x t sn

…where “t” comes from Student’s t distribution, and depends on sample size

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Student’s t Distribution vs. Normal Z Distribution

-5 0 5

0.0

0.1

0.2

0.3

0.4

Value

dens

ity

T-distribution and Standard Normal Z distribution

T with 5 d.f.

Z distribution

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Compare t and Z Values

Confidencelevel

t value with5 d.f

Z value

90% 2.015 1.6595% 2.571 1.9699% 4.032 2.58

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What happens as sample gets larger?

-5 0 5

0.0

0.1

0.2

0.3

0.4

Value

dens

ity

T-distribution and Standard Normal Z distribution

Z distribution

T with 60 d.f.

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What happens to CI as sample gets larger?

nsZx

nstx

For large samples

Z and t values become almost identical, so CIs are almost identical

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First, graph and review data Use box plot add-in Evaluate spread Evaluate how far apart mean

and median are (assume sampling design and

QC are good)

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Excel Summary Stats

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N=77

0

5

10

15

20

25

30

35

40

Ft Smith

Min 0.125th 7.5

Median 13.7

75th 18.1Max 37.9

Mean 14.8SD 8.7

1.Use the box-plot add-in

2.Calculate summary stats

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Our Question

Can we be 95%, 90%, or how confident that this mean of 14.78 is really greater than standard of 12?

We saw that N = 77, and mean and median not too different

Use z (normal) rather than t

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The mean is 14.8 +- what? We know equation for CI is

Width of confidence interval represents how sure we want to be that this CI includes true mean

Now, decide how confident we want to be

nsZx

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CI Calculation

For 95%, z = 1.96 (often rounded to 2) Stnd error (sigma/N) = (8.66/square root of

77) = 0.98 CI around mean = 2 x 0.98 We can be 95% sure that mean is included

in (mean +- 2), or 14.8-2 at low end, to 14.8 + 2 at high end

This does NOT include 12 !

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Excel can also calculate a confidence interval around the mean

Mean, plus and minus 1.93, is a 95% confidence interval that does NOT include 12!

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We know we are more than 95% confident, but how confident can we

be that Ft Smith mean > 12? Calculate where on curve our mean of 14.8 is,

in terms of z (normal) score… …or if N small, use t score

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To find where we are on the curve, calc the test statistic…

Ft Smith mean = 14.8, sigma =8.66, N =77

Calculate test statistic, in this case the z factor (we decided we can use the z rather than the t distribution)

If N was < 60, test stat is t, but calculated the same way

N

xz )(

Data’s mean

Standard of 12

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Calculate z Easily

Our mean 14.8 minus standard of 12 (treat real mean (mu) as standard) is numerator (= 2.8)

Standard error is sigma/square root of N = 0.98 (same as for CI)

so z = (2.8)/0.98 = z = 2.84 So where is this z on the curve? Remember, at z = 3 we are to the right of ~

99%

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Where on the curve?

Z = 3

Z = 2

So between 95 and 99% probable that the true mean will not include 12

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You can calculate exactly where on the curve, using Excel

Use Normsdist function, with z

If z (or t) = 2.84, in Excel

Yields 99.8% probability that the true mean does NOT include 12