Chapter 4 Exploring Chemical Analysis, Harris Statistics

4 – 1; The Gaussian Distribution Also known as a Normal Distribution Continuous probability distribution Bell Curve

4 – 1; The Gaussian Distribution Mean Locates the center of distribution Written as (pronounce x bar) Also known as the “Average” Sum of the measurements divided by the number of measurements

4 – 1; The Gaussian Distribution Variance is the square of the standard deviation (s2). Variance measures how far a set of numbers is spread out. Variance is always non-negative: a small variance indicates that the data points tend to be very close to the mean (expected value) and hence to each other, while a high variance indicates that the data points are very spread out around the mean and from each other. A variance of zero indicates that all the values are identical.

4 – 1; The Gaussian Distribution Standard Deviation An equivalent measure is the square root of the variance, called the standard deviation. Measures the width of distribution. Written as “s” The smaller the standard deviation, the narrower the results

4 – 1; The Gaussian Distribution Relative Standard Deviation Standard Deviation divided by Mean Expressed as a percentage (%)

4 – 1; The Gaussian Distribution Example on page 85 Find mean, standard deviation, and relative standard deviation for the set of measurements (7,18,10,15) Try it out using a calculator. (Store Functions)

4 – 1 The Gaussian Distribution Excel = AVERAGE () … takes the average =Stdev() … takes the standard deviation

4 – 2; Confidence Intervals Student’s t Statistical tool used to express confidence intervals to compare results from different experiments. The confidence interval is a range of values within which there is a specified probability of finding the population mean.

4 – 2; Confidence Intervals From a limited number of measurements, it’s impossible to find, ,the population mean, or the population standard deviation, . What we can determine ,sample mean and s, sample standard deviation. We say that the population mean, , is likely to lie within a certain distance from the measured mean, .

4 – 2; Confidence Intervals The range of values within which there is a specified probability of finding the true mean and true standard deviation, How certain you are that something is within a specified range.

4 – 2; Confidence Intervals S : true mean (avg) s : measured stdev n: number of measurements t : confidence level from table 4-2, Harris (table uses degrees of freedom = n – 1)

The t test 4-3; You can prove whether something is “significantly different” from something else with a certain degree of confidence. How? We compare two values of “t” We compare two sets of data

The t test 4-3; The t test for comparison of means (What is Spooled?)

The t test 4-3; If your calculated value for t is greater than the tabulated t, at the proposed confidence level, you have significant difference.

The t test 4-3; Combined “pooled” standard deviation from both sets of data

The t test

The t test 4-3; Problem 4.10

4 – 4; Outliers Data that is extremely different from a group of data 5, 6, 5, 4, 5, 6, 7, 4, 5 ,7, 898 How does this affect average? StDev? The Grubbs test: Critical values of G for rejection of outlier The method of least squares finds the “best” straight line through experimental data points.