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Uncertainty in Measurement

 p4, Figure R.3

Person Result of Measurement

1 20.15 mL

2 20.14 mL

3 20.16 mL

4 20.17 mL

5 20.16 mL

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Uncertainty in Measurement

These results show that the first three

numbers (20.1) remain the same regardless ofwho makes the measurement; these are called

 certain digits. (p5, 3)

The digit to the right of the 1 must beestimated and therefore varies; it is called an

uncertain digit.

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Reporting a Measurement

We customarily report a measurement by

recording all the certain digits plus the firstuncertain digit. (p5, 3)

In our example it would not make any sense

to try to record the volume of thousandths ofa milliliter, because the value for hundredths

of a milliliter must be estimated when using

the buret.

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Uncertainty in Measurement

 A measurement always has some degree of

uncertainty. (p5, 4)

The uncertainty of a measurement depends on

the precision of the measuring device.

Bathroom Scale Balance

Grapefruit 1 1.5 lb 1.476 lb

Grapefruit 2 1.5 lb 1.518 lb

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Significant Figures（有效數字）

Measurement = certain digits + the first

uncertain digit (the estimated number) (p5,r3)

These numbers are called the significant

figures of a measurement.

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Uncertainty in the Last Number 

The uncertainty in the last number (the

estimated number) is usually assumed to be1 unless otherwise indicated. (p5, r2)

The measurement 1.86 kilograms can be

taken to mean 1.86 0.01 kilograms.

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Precision and Accuracy

Two terms often used to describe the

reliability of measurements are precision and accuracy. (p6, 2)

Accuracy（ 準確度）refers to the agreement

of a particular value with the true value.

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Precision and Accuracy

Precision（ 精確度）refers to the degree of

agreement among several measurements ofthe same quantity. (p6, 2)

Precision reflects the reproducibility（  再現

性）of a given type of measurement.

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 p6, Figure R.4

Low accuracy Low accuracy High accuracyLow precision High precision High precision

Precision and Accuracy

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Error 

Random error（   隨機誤差）occurs in

estimating the value of the last digit of ameasurement. (p6, 3)

Systematic error（系統誤差）occurs in the

same direction each time; it is either alwayshigh or always low.

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 p6, Figure R.4

Large random Small random Small randomLarge systematic No systematic

Error 

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Average

In quantitative work, precision is often used

as an indication of accuracy.

We assume that the average of a series of

 precise measurements (which should

“average out” the random errors) is accurate,or close to the “true” value.

This assumption is valid only if systematic

errors are absent.

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Significant Figures and Calculations

Calculating the final result for an experiment

usually involves adding, subtracting,multiplying, or dividing the results of

various types of measurements. (p7, r2)

We have developed rules for counting thesignificant figures in each number and for

determining the correct number of

significant figures in the final result.

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  有效數字判斷規則（一）

 pp7-8

非零數字均為有效。

零則區分為三種：

1. 非零數字前的零僅表示小數點的位置，均為無效。0.00000025→ 二位有效。

2. 夾在有效數字間的零均為有效。1008→ 

四位有效。

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  有效數字判斷規則（二）

 p8, 1

3. 小數末尾的零皆為有效；整數末端的零   則為無效。

100→

 一位有效。

100.00→ 五位有效。

1.00 102→ 三位有效。

100.→ 三位有效。

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Exact Numbers（精確數）

 p8, 1

   計數得到，與測量無關：3  個蘋果。

   由定義得來： 1 in = 2.54 cm。

Exact Number   的有效數字視為無窮多位。

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Round Off （四捨五入）

 p9, 4

計算完成後 才對答案做四捨五入 不要

在計算過程中 對個別數字四捨五入 然

後再計算

直接由下一位有效數字決定四捨五入。

4.348  取二位有效→ 4.3。

 不是 4.348→ 4.35→ 4.4。

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  有效數字的乘除

 p9, 1

由有效數字位數最少的測量值 決定乘除

結果的有效數字位數

4.56 1.4 = 6.384→ 6.4三位 二位 二位

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  有效數字的加減

 p9, 1

小數位數最少的測量值決定加減結果的有

效數字位數

12.11 小數以下二位

18.0 一位

+) 1.013 三位

31.123→ 31.1 一位