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Blood Sugar changes-Freestyle Libre 2 vs Finger Prickle errors
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<blockquote data-quote="JohnEGreen" data-source="post: 2446706" data-attributes="member: 223921"><p>Ah yes I suppose you used the following to come to that conclusion </p><p></p><p>Repeated measurements by one instrument. Assumptions 1-3 enable us to view each measurement as a sum</p><p>Zi=μ+Xi+Y</p><p>where i is an index denoting the measurement and ranges from 1 through n.</p><p></p><p>Notice that Y has no subscript because it is a property of the instrument itself: it doesn't change from one measurement to the other. We may compute the variance of the average of the measurements--conceived of as an average of these random variables Zi--as</p><p>Var(Z¯)=1nσ2+τ2.</p><p>As n gets larger, σ2/n grows smaller.</p><p></p><p>Moreover, if we take expectations in the sense of what an arbitrarily large number of measurements would produce on average,</p><p>E[Z¯]=μ+Y</p><p>shows that even the average is biased (unless you were lucky enough to draw an instrument with Y≈0--but you can't know that).</p><p></p><p>The moral of this calculation is that averaging measurements from one instrument reduces the imprecision but has no effect on the accuracy.</p><p></p><p>Independent measurements by multiple instruments. Now i indexes both the measurement and the instrument. Accordingly,</p><p>Zi=μ+Xi+Yi.</p><p>Now</p><p>Var(Z¯)=1nσ2+1nτ2</p><p>and (in the same sense as before, taking an arbitrarily large number of instruments),</p><p>E[Z¯]=μ.</p><p>As n gets larger, both σ2/n and τ2/n grow smaller. Regardless, the expected value of the measurement is correct: Z¯ is more likely to be accurate in this case.</p><p></p><p>Thus, averaging measurements from multiple instruments reduces the imprecision and improves the accuracy.</p><p></p><p>The decision seems clear: when you have the choice, use multiple instruments. Making repeated measurements from the same instrument is no substitute.</p><p></p><p>But I say why bother just gets you very sore fingers.</p></blockquote><p></p>
[QUOTE="JohnEGreen, post: 2446706, member: 223921"] Ah yes I suppose you used the following to come to that conclusion Repeated measurements by one instrument. Assumptions 1-3 enable us to view each measurement as a sum Zi=μ+Xi+Y where i is an index denoting the measurement and ranges from 1 through n. Notice that Y has no subscript because it is a property of the instrument itself: it doesn't change from one measurement to the other. We may compute the variance of the average of the measurements--conceived of as an average of these random variables Zi--as Var(Z¯)=1nσ2+τ2. As n gets larger, σ2/n grows smaller. Moreover, if we take expectations in the sense of what an arbitrarily large number of measurements would produce on average, E[Z¯]=μ+Y shows that even the average is biased (unless you were lucky enough to draw an instrument with Y≈0--but you can't know that). The moral of this calculation is that averaging measurements from one instrument reduces the imprecision but has no effect on the accuracy. Independent measurements by multiple instruments. Now i indexes both the measurement and the instrument. Accordingly, Zi=μ+Xi+Yi. Now Var(Z¯)=1nσ2+1nτ2 and (in the same sense as before, taking an arbitrarily large number of instruments), E[Z¯]=μ. As n gets larger, both σ2/n and τ2/n grow smaller. Regardless, the expected value of the measurement is correct: Z¯ is more likely to be accurate in this case. Thus, averaging measurements from multiple instruments reduces the imprecision and improves the accuracy. The decision seems clear: when you have the choice, use multiple instruments. Making repeated measurements from the same instrument is no substitute. But I say why bother just gets you very sore fingers. [/QUOTE]
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