Significant Figures Part I: An Introduction

Objectives When you complete this presentation, you will be able to – distinguish between accuracy and precision – determine the number of significant figures there are in a measured value

Introduction Chemistry is a quantitative science. – We make measurements. mass = g length = cm width = 1.02 cm height = 3.23 cm

Introduction Chemistry is a quantitative science. – We make measurements. – We get lots of numbers. – We use those numbers to calculate things. volume = length × width × height

Introduction Chemistry is a quantitative science. – We make measurements. – We get lots of numbers. – We use those numbers to calculate things. volume = cm × 1.02 cm × 3.23 cm

Introduction Chemistry is a quantitative science. – We make measurements. – We get lots of numbers. – We use those numbers to calculate things. volume = cm 3

Introduction Chemistry is a quantitative science. – We make measurements. – We get lots of numbers. – We use those numbers to calculate things. volume = cm 3 density = mass ÷ volume

Introduction Chemistry is a quantitative science. – We make measurements. – We get lots of numbers. – We use those numbers to calculate things. volume = cm 3 density = g ÷ cm 3

Introduction Chemistry is a quantitative science. – We make measurements. – We get lots of numbers. – We use those numbers to calculate things. volume = cm 3 density = g/cm 3

Introduction Chemistry is a quantitative science. – We make measurements. – We get lots of numbers. – We use those numbers to calculate things. density = g/cm 3

Introduction Chemistry is a quantitative science. – We make measurements. – We get lots of numbers. – We use those numbers to calculate things. density = g/cm 3 – What do these numbers mean? – Do we really know the density to the nearest g/cm 3 (= 1 / 1,000,000,000 g/cm 3 )?

Accuracy and Precision Accuracy and precision are often used to mean the same thing. – We expect that an accurate measurement is a precise measurement. – Likewise, we expect that a precise measurement is an accurate measurement. They are related, but they are not the same thing.

Accuracy and Precision The accuracy of a series of measurements is how close those measurements are to the “real” value. – The “real” value of a measurement is usually the value accepted by scientists. – It is usually based on a large number of measurements made by a large number of researchers over a long period of time.

Accuracy and Precision The accuracy of a series of measurements is how close those measurements are to the “real” value. An example of accuracy is how close you come to the bullseye when shooting at a target. Accurate shots come close to the bullseye. Less accurate shots miss the bullseye.

Accuracy and Precision The precision of a series of measurements is how close the measurements are to each other. Precise shots come close to each other. Non-precise shots are not close to each other. – Which group has a greater accuracy? – The less precise group has a greater accuracy.

Accuracy and Precision When we are making a new measurement, we want to be as precise as possible. We also want to be accurate, but usually our measurement devise is already accurate. – In most cases, inaccuracy in chemistry labs is due to misreading the instrument.

Accuracy and Precision When we are making a new measurement, we want to be as precise as possible. Normally, we increase precision by making many measurements. Then, we average the measurements.

Accuracy and Precision Example 1: – Ahab determines the density of a metal several times. – His measurements are: 7.65 g/cm 3, 7.62 g/cm 3, 7.66 g/cm 3, and 7.63 g/cm 3. – He reports his average density as 7.64 g/cm 3.

Accuracy and Precision Example 1: – Brunhilda determines the density of a metal several times. – Her measurements are: 7.82 g/cm 3, 8.02 g/cm 3, 7.78 g/cm 3, and 7.74 g/cm 3. – She reports her average density as 7.84 g/cm 3.

Accuracy and Precision Example 1: Who is the most accurate? Accuracy is related to how close you are to the accepted value. MeasurementAhabBrunhilda g/cm g/cm g/cm g/cm g/cm g/cm g/cm g/cm 3 Average7.64 g/cm g/cm 3 The accepted value is 7.84 g/cm 3.

Accuracy and Precision Example 1: Who is the most accurate? Ahab’s data gives a value 0.20 g/cm 3 from the accepted value. MeasurementAhabBrunhilda g/cm g/cm g/cm g/cm g/cm g/cm g/cm g/cm 3 Average7.64 g/cm g/cm 3 The accepted value is 7.84 g/cm 3.

Accuracy and Precision Example 1: Who is the most accurate? Brunhilda’s data gives a value the same as the accepted value. MeasurementAhabBrunhilda g/cm g/cm g/cm g/cm g/cm g/cm g/cm g/cm 3 Average7.64 g/cm g/cm 3 The accepted value is 7.84 g/cm 3.

Accuracy and Precision Example 1: Who is the most accurate? Therefore, Brunhilda is the most accurate. MeasurementAhabBrunhilda g/cm g/cm g/cm g/cm g/cm g/cm g/cm g/cm 3 Average7.64 g/cm g/cm 3 The accepted value is 7.84 g/cm 3.

Accuracy and Precision Example 1: Who is the most precise? Ahab’s data varies from 7.62 to 7.66 g/cm 3 - a spread of 0.04 g/cm 3. MeasurementAhabBrunhilda g/cm g/cm g/cm g/cm g/cm g/cm g/cm g/cm 3 Average7.64 g/cm g/cm 3 The accepted value is 7.84 g/cm 3.

Accuracy and Precision Example 1: Who is the most precise? Brunhilda’s data varies from 7.74 to 8.02 g/cm 3 - a spread of 0.28 g/cm 3. MeasurementAhabBrunhilda g/cm g/cm g/cm g/cm g/cm g/cm g/cm g/cm 3 Average7.64 g/cm g/cm 3 The accepted value is 7.84 g/cm 3.

Accuracy and Precision Example 1: Who is the most precise? Ahab’s data was the most precise. MeasurementAhabBrunhilda g/cm g/cm g/cm g/cm g/cm g/cm g/cm g/cm 3 Average7.64 g/cm g/cm 3 The accepted value is 7.84 g/cm 3.

Significant Figures But, what does this have to do with significant figures? EVERYTHING!

Significant Figures The measurements we use in our calculations have a built-in precision. – When we find that the mass of an object is g, we are saying that we know the mass of the object to a precision of 0.01 g ( 1 / 100 g). – So, we know the mass to be (4×10) g + (7×1) g + (2×0.1) g + (8×0.01) g – We know the mass to 4 significant figures. 1234

Significant Figures The measurements we use in our calculations have a built-in precision. – When we find that the mass of an object is g, we are saying that we know the mass of the object to a precision of 0.01 g ( 1 / 100 g). – In the same way, we measure the length (14.34 cm), width (1.02 cm), and height (3.23 cm) to a precision of 0.01 cm. – We know the length to 4 significant figures.

Significant Figures The measurements we use in our calculations have a built-in precision. – When we find that the mass of an object is g, we are saying that we know the mass of the object to a precision of 0.01 g ( 1 / 100 g). – In the same way, we measure the length (14.34 cm), width (1.02 cm), and height (3.23 cm) to a precision of 0.01 cm. – We know the width to 3 significant figures.

Significant Figures The measurements we use in our calculations have a built-in precision. – When we find that the mass of an object is g, we are saying that we know the mass of the object to a precision of 0.01 g ( 1 / 100 g). – In the same way, we measure the length (14.34 cm), width (1.02 cm), and height (3.23 cm) to a precision of 0.01 cm. – We know the height to 3 significant figures.

Significant Figures It should be simple to tell how many significant figures there are in a measurement. For example: if we measure the length of the room to be 14 meters, we have 2 significant figures. But 14 m = 1400 cm – 1400 cm has 4 digits – It still has only 2 significant figures.

Significant Figures It should be simple to tell how many significant figures there are in a measurement. For example: if we measure the length of the room to be 14 meters, we have 2 significant figures. And, 14 m = km – has 4 digits – It still has only 2 significant figures.

Significant Figures We have rules for determining the number of significant figures in a measurement. There is an easy way to determine the number of significant figures in a measurement. – We convert the number to scientific notation, and count the number of significant figures. 450,000 = 4.5 × 10 5 ➠ 2 significant figures

Significant Figures We have rules for determining the number of significant figures in a measurement. There is an easy way to determine the number of significant figures in a measurement. – We convert the number to scientific notation, and count the number of significant figures = × 10 −2 ➠ 4 significant figures

Significant Figures We have rules for determining the number of significant figures in a measurement. There is an easy way to determine the number of significant figures in a measurement. – We convert the number to scientific notation, and count the number of significant figures. 14 = 1.4 × 10 1 ➠ 2 significant figures

Significant Figures We have rules for determining the number of significant figures in a measurement. There is an easy way to determine the number of significant figures in a measurement. – We convert the number to scientific notation, and count the number of significant figures. 1,400 = 1.4 × 10 3 ➠ 2 significant figures

Significant Figures We have rules for determining the number of significant figures in a measurement. There is an easy way to determine the number of significant figures in a measurement. – We convert the number to scientific notation, and count the number of significant figures = 1.4 × ➠ 2 significant figures

Significant Figures We have rules for determining the number of significant figures in a measurement. There is an easy way to determine the number of significant figures in a measurement. – We convert the number to scientific notation, and count the number of significant figures = 1.30 × 10 1 ➠ 3 significant figures

Significant Figures We have rules for determining the number of significant figures in a measurement. There is an easy way to determine the number of significant figures in a measurement. – We convert the number to scientific notation, and count the number of significant figures = × ➠ 4 significant figures

Examples How many significant figures are in each of the following numbers? 1.4,210 m s 3.5,100,000 kg mL cm L 4.21×10 3 m ➠ 3 significant figures 2.543×10 -4 s ➠ 4 significant figures 5.1×10 6 kg ➠ 2 significant figures 7.45×10 -1 mL ➠ 3 significant figures 4.324×10 0 cm ➠ 4 significant figures 7.00×10 -3 L ➠ 3 significant figures

Summary Accuracy relates to how close a value is to an accepted value. Precision relates to how close individual measurements are to each other. Significant figures are a measure of the precision of our measurements.