Section 2.2 ~ Dealing With Errors. Introduction to Probability and Statistics Fall 2015

Transcription

1 Section 2.2 ~ Dealing With Errors Introduction to Probability and Statistics Fall 2015

2 Objective To understand the difference between random and systematic errors, be able to describe errors by their absolute and relative sizes, and know the difference between accuracy and precision in measurements.

3 Types of Error Broadly speaking, measurement errors fall into two categories: random errors and systematic errors Random errors occur because of random and inherently unpredictable events in the measurement process Examples ~ weighing a baby that is shaking the scale Copying the measurement down wrong Reading a measuring device wrong Systematic errors occur when there is a problem in the measurement system that affects all measurements in the same way Examples ~ An error in the calibration of any measuring device; A scale that reads 1.2 pounds with nothing on it A clock that is 5 minutes slow

4 How to deal with these errors Random errors can be minimized by taking many measurements and averaging them Systematic errors are easy to fix when discovered, you can go back and adjust the measurements accordingly

5 Example 1 Scientists studying global warming need to know how the average temperature of the entire Earth, or the global average temperature, has changed with time. Consider two difficulties in trying to interpret historical temperature data from the early 20 th century: (1) Temperatures were measured with simple thermometers and the data were recorded by hand, and (2) most temperature measurements were recorded in or near urban areas, which tend to be warmer than surrounding rural areas because of heat released by human activity. Discuss whether each of these two difficulties produces random or systematic errors, and consider the implications of these errors. The first difficulty would most likely involve random errors because people undoubtedly made errors in reading the thermometer and recording the data The second difficultly would be an example of a systematic error since the excess heat in the urban areas would always cause the temperature to be higher than it would be otherwise.

6 Size of Errors: Absolute versus Relative Is the error big enough to be of concern or small enough to be unimportant? Scenario: Suppose you go to the grocery store and buy what you think is 6 pounds of hamburger, but because the store s scale is poorly calibrated you actually get only 4 pounds. You d probably be upset by this 2 pound error. Now suppose that you are buying hamburger for a huge town barbeque and you order 3000 pounds but only receive 2998 pounds. You are short by the same 2 pounds as before, but in this case the error probably doesn t seem as important. The size of an error can differ depending on how you look at it: Absolute error describes how far the claimed or measured value lies from the true value Example ~ the 2-pound error on the scale at the grocery store Relative error compares the size of the absolute error to the true value and is often expressed as a percentage Example ~ the case of buying only 4 pounds of meat because of the 2 pound error on the scale would result in a 50% relative error since the absolute error of 2 pounds is half the actual weight of 4 pounds

7 Absolute Error Absolute error = claimed or measured value - actual value Example 2: a. Your true weight is 100 pounds, but a scale says you weight 105 pounds. Find the absolute error. Absolute error = 105 lb lb Absolute error = 5 lb The measured weight is too high by 5 pounds b. The government claims that a program costs $99.0 billion and the true cost is $100.0 billion. Find the absolute error. Absolute error = $99.0 billion - $100.0 billion Absolute error = - $1.0 billion The claimed cost is too low by $1.0 billion A positive absolute error will occur when the measured value is higher than the true value A negative absolute error will occur when the measured value is lower than the true value

8 Relative Error Absolute error claimed or measured value - actual value Relative error = 100% actual value Example 3: a. Your true weight is 100 pounds, but a scale says you weigh 105 pounds. Find the relative error. claimed or measured value - actual value Relative error = 100% actual value 105 lb- 100 lb Relative error = 100% 100 lb 5 lb Relative error = 100% 100 lb Relative error = 5% Since the measured value was higher than the true value, the relative error is positive. The measured weight was too high by 5%.

9 Relative Error Absolute error measured value - actual value Relative error = 100% actual value Example 3: b. The government claims that a program costs $99.0 billion and the true cost is $100.0 billion. Find the relative error. claimed or measured value - actual value Relative error = 100% actual value $99.0 billion - $100.0 billion Relative error = 100% $100.0 billion -$1.0 billion Relative error = 100% $100.0 billion Relative error = -1% Since the measured value was lower than the true value, the relative error is negative. The claimed cost was too low by 1%.

10 Relative Error A positive absolute error will occur when the measured value is higher than the true value This forces the RELATIVE ERROR to be positive as well! A negative absolute error will occur when the measured value is lower than the true value This forces the RELATIVE ERROR to be negative as well!

11 Practice You and your friend are laying out a fence behind your house. You use a tape measure to determine measurements. You determine that you will need 200 feet of fencing to enclose the area you laid out. Name one source of likely errors: Will those errors be random or systematic? If the actual amount of fencing you need is 205 feet, what is the absolute error? What is the relative error?

12 Describing Results: Accuracy and Precision Once a measurement is reported, we can evaluate it in terms of its accuracy and precision Accuracy describes how close a measurement lies to the true value Example ~ A census count was 72,453 people, but the true population was 96,000 people. Not very accurate because it is nearly 25% smaller than the actual population Precision describes the amount of detail in a measurement Example ~ census; the value 72,453 is very precise as it seems to tell us the exact count as opposed to an estimate like 72,400

13 Example 4 Suppose that your true weight is pounds. The scale at the doctor s office, which can be read only to the nearest quarter pound, says that you weigh 102¼ pounds. The scale at the gym, which gives a digital readout to the nearest 0.1 pound, says that you weigh pounds. Which scale is more precise? Which is more accurate? The scale at the gym is more precise because it gives your weight to the nearest tenth of a pound as opposed to the nearest quarter of a pound. The scale at the doctor s office is more accurate because its value is closer to your true weight.

14 Summary Two basic types of errors: random and systematic The size of an error can be described as either absolute or relative Once a measurement is reported, it can be evaluated in terms of its accuracy and precision

15 Homework Pg #1-4 #9-12 #16 #21-24 #27-28