Understanding Accuracy, Precision and Uncertainties in Measurements, Summaries of Elementary Mathematics

Download Understanding Accuracy, Precision and Uncertainties in Measurements and more Summaries Elementary Mathematics in PDF only on Docsity! Accuracy, Precision and Uncertainties Precision • How close your results are to each other • How repeatable they are • The number of decimal places or significant figures that an instrument can measure Example 2 • The actual temperature of a classroom is 18.5°C • A digital thermometer measures it to be 19°C - is this accurate or precise? It is accurate because it gives a value close to the actual value, but it is not very precise because it only reads to two significant figures Random errors • Human errors e.g. reaction time when using a stopwatch or parallax errors when reading a graduated scale • Unpredictable environmental changes e.g. wind blowing during an outdoor experiment to measure the speed of sound • Random errors can be reduced by repeating the experiment and averaging your results • Random errors mainly affect precision Systematic errors • Equipment errors e.g. digital scales being incorrectly zeroed or a thermometer that consistently reads 3°C too high • Systematic errors can be reduced by replacing equipment, calibrating equipment or using multiple sets of equipment to compare readings • Systematic errors mainly affect accuracy Percentage difference • The percentage difference is a simple calculation to compare the value obtained in an experiment to its true value • It gives an idea of how accurate your result is % 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑒𝑒𝑒𝑒𝑑𝑑𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑𝑑𝑑𝑒𝑒𝑒𝑒𝑒𝑒 𝑣𝑣𝑒𝑒𝑒𝑒𝑣𝑣𝑑𝑑 − 𝑒𝑒𝑑𝑑𝑣𝑣𝑑𝑑 𝑣𝑣𝑒𝑒𝑒𝑒𝑣𝑣𝑑𝑑 𝑒𝑒𝑑𝑑𝑣𝑣𝑑𝑑 𝑣𝑣𝑒𝑒𝑒𝑒𝑣𝑣𝑑𝑑 × 100 % • The vertical bars mean take the ‘modulus’ (absolute value) – so you ignore the minus sign if the numerator is negative • A rule of thumb is that if your % difference is less than 10% then your experimental result is fairly accurate Example 1 • An experiment to measure the time taken to travel a certain distance gives a reading of 11.8 s. The true time taken is 10.3 s. Calculate the percentage difference and comment on the accuracy of the experiment. % 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 11.8−10.3 10.3 × 100 % = 14.6 % It is not a very accurate experiment because the % difference is greater than 10 %. Example 2 • An experiment to measure the density of steel gives a value of 7895 kgm-3. The true value is 8050 kgm-3. Calculate the percentage difference and comment on the accuracy of the experiment. % 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 7895−8050 8050 × 100 % = 1.9 % It is a very accurate experiment because the % difference is much less than 10 %. Summary questions 2 - answers Question 2 % 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 1.01−1.06 1.06 × 100 % = 4.7 % It is an accurate experiment because the % difference is less than 10 %. Absolute uncertainties • These are expressed as ± the smallest increment that your instrument or measuring device can measure. • For example a stopwatch that can measure to 0.1 s has an absolute uncertainty of ± 0.1 s. • So, if this stopwatch gives a reading of 5.0 s it should really be quoted as 5.0 s ± 0.1 s. • This means that the range of true values is 4.9 s to 5.1 s. Percentage uncertainties • These are the absolute uncertainties expressed as a percentage of the measured value • So, for the stopwatch that produces a reading of 5.0 s ± 0.1 s, the percentage uncertainty is: 0.1 5.0 × 100 = 2 % Absolute and Percentage uncertainties • This is why in an experiment you want your measured range to be as large as possible – because it reduces the % uncertainty in your result. • For example measuring the period of a pendulum oscillation by timing over multiple swings rather than just one swing Absolute and Percentage uncertainties • At A Level, you need to be able to calculate the percentage uncertainty of a quantity derived from a formula - for example, the percentage uncertainty in speed from measured values of distance and time. • To work out the percentage uncertainty of a quantity derived from a formula, the uncertainties from the measured values MUST BE EXPRESSED AS PERCENTAGE UNCERTAINTIES FIRST – this is usually the first step in these questions. • If the measured quantities in the formula are being multiplied or divided YOU ALWAYS ADD THE PERCENTAGE UNCERTAINTIES. • If you have a squared (or cubed) quantity then you ADD the uncertainties just the same as if it were any other two quantities being multiplied together – TAKE CARE HERE, MANY A LEVEL PUPILS LOSE MARKS WITH THIS. Example 1 A runner runs in a race. The distance is measured as 250 m ± 1 m and the time is measured as 31 s ± 0.1 s. Calculate the percentage uncertainty in their average speed. Step 1: Work out the percentage uncertainties of distance and time Distance = 1/250 x 100 = 0.40 % Time = 0.1/31 x 100 = 0.32 % Step 2: Look at the formula for average speed: average speed = distance/time Distance and time are divided – this means that to calculate the % uncertainty in speed, you ADD the % uncertainties in distance and time. Step 3: Percentage uncertainty in average speed = 0.40 + 0.32 = 0.72% Summary questions 3 Please write your answers in your book and attempt these questions without looking back at the previous slides. Spend 10 minutes on this. 1. A cyclist travels 3500 m ± 60 m in a time of 1200 s ± 30 s. Calculate the percentage uncertainty in their average speed. 2. An object with a mass of 80 kg ± 2 kg travels with a velocity of 250 m/s ± 5 m/s. Calculate the percentage uncertainty in its kinetic energy. Summary questions 3 - answers Question 1 A cyclist travels a distance of 3500 m ± 60 m in a time of 1200 s ± 30 s. Calculate the percentage uncertainty in their average speed. Step 1: Work out the percentage uncertainties of distance and time Distance = 60/3500 x 100 = 1.7 % Time = 30/1200 x 100 = 2.5 % Step 2: Look at the formula for average speed: average speed = distance/time Step 3: Percentage uncertainty in average speed = 1.7 + 2.5 = 4.2 % Summary questions 3 - answers Question 2 An object with a mass of 80 kg ± 2 kg travels with a velocity of 250 m/s ± 5 m/s. Calculate the percentage uncertainty in its kinetic energy. Step 1: Work out the percentage uncertainties of mass and velocity Mass = 2/80 x 100 = 2.5 % Velocity = 5/250 x 100 = 2.0 % Step 2: Look at the formula for kinetic energy: kinetic energy = 0.5 x mass x velocity2 Step 3: Percentage uncertainty in kinetic energy = 2.5 + 2.0 + 2.0= 6.5 %