standard deviation and standard error
Exercises
(2.1) Mean, standard deviation and standard error (I ) l
An expenment was conducted to determine the concen- and fourth columns should be a x . /2 N .. 2
tration of a sodium hydroxide solution. The eight repeat 1 l t’ l A we e’ te
measurements of the volume of hydrochloric acid titrated anda x + m ,respec we y. s at m r-
(all in ml) are: 25.8, 26.2, 26.0, 26.5, 25.8. 26.1, 25.8 and ested in the variation over a large dynamic range, choose
26.3. Calculate (i) the mean. (ii) the standard deviation values for N such as 2, 3, 5, 10, 20. 30, etc. Verify the
using the rough-and-ready approach; (iii) the standard statement from Section 2.7.] that the number of data
deviation using eqn (2.3); (iv) the standard error of the points, N , needs to approach a few tens of thousands
volume. before the second significant figure in the error can be
(2 2) Mean standard deviation and st l rd error (2) quoted, i.e. when the values in the three columns become
equal to the second significant figure. Repeat the analy-
12. measurements Of the sensitmty Of a photodiode C”. sis for the case where a = 0 123 456 789 ie the first
2‘32“: gpglgjn; 32’; 23:33 :n315’540913ga?08ul:t:5(,i:3: Significant digit of the error is 1. How many data points
mean. (ii) the standard deviation using eqn (23); (iii) the Lneusltlig‘ollected before the third Significant figure can
standard error. q
(2.5) Reporting results (1)
(2’3) gegfigggefgdfeggfifnzgrafiera minute of collect- Fifteen measurements of a resistance are quoted here.
ing data the statistical noise was reduced to l picotesla. 33:? gntzzgmggatgz égerepi‘: measurerlrzients‘} 0313′
For how much longer should data be collected in order mistakes in the othy it go en ru es. 3″” y e
to reduce the random error by a factor of 10? ‘ er resu 5’
(1) (99.8 i 0.270) x 103 o,
(2.4) Error in the error ,, 3
Consider a set of measurements with the standard error (it) (100 i 0’3) x 10 52’
calculated to be a = 0.987 654 321. Here we address (Ill) (1000:}: 0.3) X 103 $2,
the question of how many significant figures should be (iv) (100.! :l: 0.3) x 1039
quoted. Construct a spreadsheet with four columns. The (v) 97. l x lO3:l:276 S2
first column should be N, the number of measurements
on which a is based. In the second column write a (1″) (998645 530-2701) X 103 9.
to the nine significant figures quoted above. The third (Vii) 98.6 x 103 :l: 3 x 102 $2,
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