Section 2.2 ~ Dealing With Errors

Introduction to Probability and StatisticsMs. Young

Objective

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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.

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

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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

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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 20th 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 would always cause the temperature to be higher than it would be otherwise.

Refer to “The Census” case study on p.61 for another example

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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

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Absolute Error

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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.

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.

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

Absolute error = 105 lb - 100 lb

Absolute error = 5 lb

Absolute error = $99.0 billion - $100.0 billion

Absolute error = - $1.0 billion

Relative Error

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claimed or measured value - actual valueRelative 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.

Since the measured value was higher than the true value, the relative error is positive. The measured weight was too high by 5%.

Absolute error

claimed or measured value - actual valueRelative error = 100%

actual value

105 lb- 100 lbRelative error = 100%

100 lb

5 lbRelative error = 100%

100 lb

Relative error = 5%

Relative Error

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measured value - actual valueRelative 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.

Since the measured value was lower than the true value, the relative error is negative. The claimed cost was too low by 1%.

Absolute error

claimed or measured value - actual valueRelative error = 100%

actual value

$99.0 billion - $100.0 billionRelative error = 100%

$100.0 billion

-$1.0 billionRelative error = 100%

$100.0 billion

Relative error = -1%

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

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Example 4 Suppose that your true weight is 102.4 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 100.7 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.

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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

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