CHEMISTRY ANALYTICAL CHEM 3811
CHAPTERS 4
DR. AUGUSTINE OFORI AGYEMAN
Assistant professor of chemistry Department of natural sciences Clayton state university
CHAPTER 4 STATISTICS
STATISTICS
- Random errors are always associated with measurements - No conclusion can be drawn with complete certainty - Scientists use statistics to accept conclusions that have high probability of being correct and to reject conclusions that have low probability of being correct - Statistics deals with only random errors - Systematic errors should be detected and eliminated
THE GAUSSIAN DISTRIBUTION
- Symmetric bell shaped curve representing the distribution of experimenal data - Characterized by mean and standard deviation - The Gaussian function is
f(x) = ae
(x ) 2 2 2
1 - a is the height of the curves peak, a = 2
- is the position of the center of the peak (the mean) - is a measure of the width of the curve (standard deviation)
THE GAUSSIAN DISTRIBUTION
THE GAUSSIAN DISTRIBUTION
Arithmetic Mean - Also known as the average - Is the sum of the measured values divided by the number of measurements
_
x =
x
i
i
n
1 = ( x 1 + x 2 + x 3 +..... + x n ) n
= sigma, symbol for the sum xi = a measured value n = number of measurements
THE GAUSSIAN DISTRIBUTION
Standard Deviation - A measure of the width of the distribution - Small standard deviation gives narrow distribution curve
s =
x i x i n 1
_
2
xi = a measured value n = number of measurements n-1 = the degrees of freedom
Standard deviation has the same number of decimal places as the mean
THE GAUSSIAN DISTRIBUTION
Relative Standard Deviation
Relative Standard Deviation =
s x
_
x 100
- The sample mean and sample standard deviation (x and s) apply to finite set of measurements - For infinite set of data use is made of the true mean or population mean (designated ) and the true standard deviation (designated )
THE GAUSSIAN DISTRIBUTION
Variance - Is the square of the standard deviation - Variance = 2
THE GAUSSIAN DISTRIBUTION
Median - The middle number in a series of measurements arranged in increasing order - The average of the two middle numbers if the number of measurements is even
THE GAUSSIAN DISTRIBUTION
Mode - The value that occurs the most frequently Range - The difference between the highest and the lowest values All but the range should be expressed with one extra digit beyond the last significant digit
THE GAUSSIAN DISTRIBUTION
Probability - Range of measurements for ideal Gaussian distribution - The percentage of measurements lying within the range given (one, two, or three standard deviation on either side of the mean) Range 1 2 3 Gaussian Distribution (%) 68.3 95.5 99.7
CONFIDENCE INTERVAL
- Range of values within which there is a specified probability of finding the true mean () - The t-test is used to express confidence intervals - Also used to compare results from different experiments
CONFIDENCE INTERVAL
ts = x n
_
- That is, the range of confidence interval is ts/n below the mean and + ts/n above the mean - For better precision reduce confidence interval by increasing number of measurements Refer to table 4-2 on page 87 for t-test values
CONFIDENCE INTERVAL
To test for comparison of Means - Calculate the pooled standard deviation (spooled) - Calculate t - Compare the calculated t to the value of t from the table - The two results are significantly different if the calculated t is greater than the tabulated t at 95% confidence level
CONFIDENCE INTERVAL
For two sets of data with - n1 and n2 measurements - averages of x1 and x2 - standard deviations of s1 and s2
s pooled =
2 s1 ( n1 1) + s 2 ( n 2 1) 2 n1 + n 2 2
t =
x1 x 2 s pooled
n1n 2 n 1 +n 2
Degrees of freedom = n1 + n2 - 2
GRUBBS TEST FOR AN OUTLIER
- An outlier is a datum that is far from other points - Grubbs test is used to determine whether an outlier should be rejected or retained
G=
questionable value x s
GRUBBS TEST FOR AN OUTLIER
- Calculate mean, standard deviation, and then G - Compare G calculated to G tabulated (Table 4-4 on page 92) - The questionabl...