1. Error Analysis, Statistics, Graphing and Excel
Necessary skills for Chem V01BL

2. Accuracy and Precision
In experiments there are errors. We measure how big these errors are with
Accuracy – is how close our value is to a reference (accepted value) - it is normally reported as the % error and measures systematic error
Precision – how repeatable is our result is. It is reported in terms of the standard deviation this is a measure of random error

3. Accuracy and Precision
Accurate but not precise
Random error > systematic error
Precise but not accurate
Systematic error > random error

4. Error Analysis Mean Value
Repeat measurements xi in N independent trials
Report the mean value
This equation adds up all the values and divide the sum by the number of values
Reduces random errors from multiple trials

5. Measuring Accuracy Percent Error
Accuracy: the % error measures closeness of agreement between a measured and true value.
Accuracy is a measure of systematic error (biases)

6. Measuring Precision Standard Deviation s
σ is a measure of precision. The smaller σ is, the more precise the measurement was
This normally requires at least 6 independent trials
Can be evaluated for a column of data in Excel® (or Numbers®) with the =STDEV(A1:A10) type syntax

7. Precision Continued
If an observable is subject to random error, we get a binomial (Gaussian) distribution about the mean value.
s is a measure of the width of the distribution, and therefore the magnitude of the random error
Reporting 𝑥±𝜎 describes the dark blue region (68.2% or the trials fell in this range)
Reporting 𝑥±2𝜎 include the next shade of blue as well and spans 95.4% of all trials. In Science we generally report answer in this way.
68.2% of trials
95.4% of trials
We use this

8. Random Error
Standard Deviation Continued
More precise

9. Rejecting Outliers: Q Test
The Q test is a way of deciding if any given trial is an outlier
The Q test tells us with 90%, 95% or 99% confidence that a trial point is in error and can be thrown out
If so the average and standard deviation can be recalculated without that point

10. Rejecting Outliers: Q Test
If Q > Qx% then the point is rejected at the x% confidence level

11. Rejecting Outliers: Q Testx
gap Q
0.219
0.004
0.0296
0.223
0.008
0.0593
0.231
0.015
0.111
0.246
0.354
0.108
0.8
average
0.255
0.135
range
σ = 0.057
𝑄= 0.354− −0.219 = =0.8
The average is 0.255±0.057 (68% confidence level)
Q90% = 0.642, Q95% = 0.710, and Q99% = 0.821
If Q > Qx% then the point is rejected at the x% confidence level
Q is largest for the last point (Q = 0.8 > Q95%) which means it can be rejected at the 95% level
If we reject the last value, the average becomes 0.230±0.031

12. Graphing and Excel
Graphing is used to see mathematical relationships (correlations) between a dependent and independent variable
The correlation is measured by calculating the coefficient of determination R2
𝑅 2 =1− 𝑖=1 𝑁 𝑦 𝑥 𝑖 −𝑓 𝑥 𝑖 𝑖=1 𝑁 𝑦 𝑥 𝑖 − 𝑦 2
y(xi) is the value of the independent value for the dependent value and f(xi) is the function attempting model y(xi)=f(xi)
Use Excel to plot two variables which are linearly correlated and use Excel to determine the equation for the line as well as R2
As we shall see if R2>0.99 data fits the model

13. Graphing and Excel Correlated data: ideal gas law
Find the relationship between the volume of 1 mole of O2 and its temperature at 1 atmosphere
The equation for the straight line where y = V and x = T gives
The R2=1 means there is a perfect linear correlation between T and V
V(L)
T(oC) 25
31.49 30
92.38 35
153.28 40
214.18 45
275.08 50
335.97

14. Graphing and Excel Correlated data: ideal gas law
Find the relationship between the volume of 1 mole of O2 and its Pressure at 273K. P is the independent variable (x) and Volume the dependent variable (y)
P(torr)
V(mL)
400
42.6
500
34.1
600
28.4
700
24.3
800
21.3
900
18.9
1000 17
1100
15.5
1200
14.2
𝑉 𝑚𝐿 = (𝑇𝑜𝑟𝑟.𝑚𝐿) 𝑃(𝑇𝑜𝑟𝑟)
Black line is a linear trendline y = mx + b
Orange line is a power trendline y = mxb

15. Graphing and Excel
Uncorrelated data House value versus Street Address Number

16. Group Activity Pair up with your neighbor
Measure the mass of an empty cylinder and record it on the sheet given (Part A)
Add 10 mL of DI water and re-measure the mass of the cylinder on the sheet (Part A)
Measure the temperature of the water to 0.1oC and record on the sheet
Look up and record the density for water at that temperature
Calculate the mass of the 10 mL of water (Part A)
Record your answer on the board
Copy all the board data onto your sheet in Part B
Calculate the volume for each trial from the board
Calculate the mean volume and the standard deviation
Complete the sheet