1. ANALYTICAL CHEMISTRY CHEM 3811
CHAPTER 4
DR. AUGUSTINE OFORI AGYEMAN
Assistant professor of chemistry
Department of natural sciences
Clayton state university

2. CHAPTER 4
STATISTICS

3. 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

4. THE GAUSSIAN DISTRIBUTION
- Symmetric bell shaped curve representing the distribution
of experimenal data
- Characterized by mean and standard deviation
- The Gaussian function is
- a is the height of the curve’s peak,
- µ is the position of the center of the peak (the mean)
- σ is a measure of the width of the curve (standard deviation)

5. THE GAUSSIAN DISTRIBUTION
f(x)
-3σ
-2σ -σ μ σ 2σ 3σ x

6. THE GAUSSIAN DISTRIBUTION
Arithmetic Mean
- Also known as the average
- Is the sum of the measured values divided by the number
of measurements
∑ = sigma, symbol for the sum
xi = a measured value
n = number of measurements

7. Standard deviation has the same number of decimal places as the mean
THE GAUSSIAN DISTRIBUTION
Standard Deviation
- A measure of the width of the distribution
- Small standard deviation gives narrow distribution curve
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

8. THE GAUSSIAN DISTRIBUTION
Relative Standard Deviation
- 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 µ) and
the true standard deviation (designated σ)

9. - Is the square of the standard deviation
THE GAUSSIAN DISTRIBUTION
Variance
- Is the square of the standard deviation
- Variance = σ2

10. 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

11. 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

12. THE GAUSSIAN DISTRIBUTION
Probability
- Range of measurements for ideal Gaussian distribution
- The percentage of measurements lying within the given range
(one, two, or three standard deviation on either side of the mean)
Range
µ ± 1σ
µ ± 2σ
µ ± 3σ
Gaussian Distribution (%)
68.3
95.5
99.7

13. CONFIDENCE INTERVAL
- Range of values within which there is a specified probability
of finding the true mean (µ)
- The t-test is used to express confidence intervals
- Also used to compare results from different experiments

14. CONFIDENCE INTERVAL - 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

15. To test for comparison of Means
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

16. For two sets of data with
CONFIDENCE INTERVAL
For two sets of data with
- n1 and n2 measurements
- averages of x1 and x2
- standard deviations of s1 and s2
Degrees of freedom = n1 + n2 - 2

17. 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

18. 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 questionable datum is rejected if G calculated is greater than
G tabulated
- The questionable datum is retained if G calculated is smaller than

19. (METHOD OF LEAST SQUARES)
BEST STRAIGHT LINE
(METHOD OF LEAST SQUARES)
The equation of a straight line
y = mx + b
m is the slope (y/x)
b is the y-intercept (where the line crosses the y-axis)

20. BEST STRAIGHT LINE (METHOD OF LEAST SQUARES)
The method of least squares
- finds the best straight line
- adjusts the line to minimize the vertical deviations
Only vertical deviations are adjusted because
- experimental uncertainties in y values > in x values
- calculations for minimizing vertical deviations are easier

21. BEST STRAIGHT LINE (METHOD OF LEAST SQUARES)
- n is the number of data points
Knowing m and b, the equation of the best straight line can
determined and the best straight line can be constructed

22. (METHOD OF LEAST SQUARES)
BEST STRAIGHT LINE
(METHOD OF LEAST SQUARES)
xiyi
∑(xiyi) =
xi2
∑xi2 = xi
∑xi = yi
∑yi =