In this mini-lesson, we shall explore the topic of accuracy,* *by finding answers to questions like what is meant by accuracy, what is the difference between accuracy and precision, and accuracy on the number line.

**Understanding Accuracy and Precision**

Quite often we observe people taking a train to work and set their schedules based on the time of the trains.

Let us assume that we have a daily train at a standard time of 7.00 AM. Observe the below timing of the train across two weeks.

Timings | |

Week - I | 7:01 AM, 7:03AM, 7:00AM, 7:01AM, 7:03AM |

Week - II | 7:02AM, 7:02AM, 7:02AM, 7:02AM, 7:02AM |

For analyzing these above timings, the concepts of accuracy and precision are very helpful.

Let's move ahead and get to know more about accuracy and precision.

**Lesson Plan**

**What Is Meant by Accuracy?**

Accuracy measures the closeness of the measurement to the actual value. Accuracy is a measure of the quality of measurement.

In the process of purchase of a pound of bread from a supermarket, if the measured weight of 1 pound is equal to the printed weight of 1 pound, then we can call this measurement to be accurate.

Accuracy is not directly calculated but can be calculated based on the errors. A measurement with low errors represents high accuracy.

Further, to understand more about the accuracy, let us look into the formulae for error calculations.

- Absolute Error (Error): This is the difference between the actual value and the experimental value of the data.

\[Absolute Error \\= Experimental Value - Actual Value\] - Relative Error: This is the ratio of the error in data and the actual data value. The error in the difference between the experimental value and the actual value. \[\text{Relative Error} \\= \dfrac{ \text{(Experimental Value - Actual Value)}}{\text{Actual Value}} \]
- Percentage Error: Here the above relative error is expressed in percentage. \[ Percentage Error \\= Relative Error \times 100 \]

**Explain Accuracy on a Number Line **

First, let us understand the "Number line".

A number line is a straight line along with numbers placed at equal intervals. A number line can also be with negative and positive integers having zero in the middle.

All right! How do we check accuracy on a number line?

The above figure shows the points of accuracy near to the true value, but the precision points are near to each other but are far from the true value.

Let us understand this with an example.

Two students when asked to divide a number 10 with 4. One gives an answer as 2.5 and another student gives 2.4678 as the answer. The first student is said to be accurate and the second student can be said to be precise.

**What Is the Difference Between Accuracy and Precision?**

Generally, accuracy and precision are considered synonyms. But from a scientific and engineering perspective, they are words with a different meaning.

Let us check the following points, to understand the difference between accuracy and precision.

Accuracy | Precision |
---|---|

Accuracy is the closeness of the measured value to the actual value. | Precision is the closeness of one measurement with another measurement. |

Accuracy can be low due to human error. | Precision can be attributed to errors in the instrument being used. |

A single measurement is sufficient to know the accuracy. | A series of measurements are required to define precision. |

Example: The measured height of a wall as 8.1 feet when the actual height is 8 feet, is an example of accuracy. |
Example: A pressure gauge (used to measure atmospheric pressure) which gives readings as 1.0813 atm, 1.0815 atm, and 1.0814 atm, is an example of precision. |

**Why Is Accuracy Important?**

Accuracy is needed to know the deviation from the actual value. This will help us to decide if a given measurement can be acceptable or not.

Generally, a range is defined with the true value as the midpoint of the range. All the values which fall within this defined range are considered as accurate.

As an example, consider the level of temperature required to cook a pizza. The oven temperature should be 400 degrees for the pizza to be cooked well.

But for practical reasons, the accurate temperature of 400 degrees is not always possible.

Hence a range is defined from 390 degrees to 410 degrees, with the 400-degree mark as the midpoint of the range. The temperatures falling within this range, are considered as accurate.

- A team of scientists is trying to measure the depth of a sea. The team uses a depth finder and measures the depth of the sea as 23,450 feet. Also the depth finder has an error rate of \( +0.01 \% \). Calculate the actual depth of the sea.

**Solved Examples **

Example 1 |

John used a thermometer to measure the temperature in the laboratory. The thermometer shows a temperature of 43.5 degrees Fahrenheit when the actual temperature is 45 degrees Fahrenheit. Help John to find the Absolute Error, Relative Error, and Percentage Error.

**Solution**

\(\begin{align*} \text {Actual Reading} &= 45 \text{ Degrees Fahrenheit}\\ \text {Observed Reading} &= 43.5 \text{ Degrees Fahrenheit} \end{align*} \)

\[\begin{align*} \text {Absolute Error} &= \text {Actual Reading - Observed Reading}\\ &= 45 - 43.5\\ &= {1.5 \text{ Degrees Fahrenheit}} \end{align*} \]

\[\begin{align*} \text {Relative Error} &= \dfrac{\text {(Actual Reading - Observed Reading)}}{\text{Actual Reading}}\\ &= \dfrac{(45 - 43.5)}{45}\\ &= \dfrac{1.5}{45}\\ &= 0.033 \end{align*} \]

\[\begin{align*} \text {Percentage Error} &= \text{Relative Error} \times 100\\ &= 0.033 \times 100\\ &= 33\% \end{align*} \]

\(\therefore\) The absolute error is 1.5, the relative error is 0.033, and the percentage error is 3.33% |

Example 2 |

Describe the level of accuracy and precision of the markings, on each of the given dartboards.

(a)

(b)

(c)

(d)

**Solution**

(a) All the points are away from the target, but the points are clustered together. This represents **high precision and low accuracy**.

(b) Here the points are distributed and none of the points are near to the target. This is a case of **low precision and low accuracy**.

(c) The points are all clustered together and they fall near the target. This signifies **high precision and high accuracy**.

(d) The points are all distributed but are all near to the target. This is a case of **low precision and high accuracy. **

\(\therefore\) The four situations represent different levels of accuracy and precision. |

Example 3 |

Sam has a clock that loses 10 minutes every 24 hours. On a Sunday at 9 AM, Sam has set the time. What will be the time shown by this watch on Wednesday at 9 PM?

**Solution**

Given that the clock loses 10 minutes every 24 hours.

The time duration between Sunday 9 AM and Wednesday 9 PM is 3.5 days.

Loss of time = 3.5 x 10 minutes

= 35 minutes

The time shown by the clock on Wednesday 9 PM = 9.00 + 35 minutes

= 9.35 PM

\(\therefore\) The correct time on Wednesday is 9.35 PM |

**Interactive Questions**

**Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

The mini-lesson targeted the fascinating concept of accuracy. The math journey around accuracy starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

**About Cuemath**

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

**FAQs on Accuracy**

### 1. How do you define accuracy?

Accuracy is the closeness of the measurement with the actual value.

### 2. What is accuracy in your own words?

Any measurement which is exactly equal to the defined value can be said as accurate.

### 3. How to calculate accuracy?

Accuracy is not directly calculated but is only understood in terms of error. Errors can be measured and calculated.

Error Rate (Percentage Error) = \(\dfrac{(\text{Measured Value - True Value})}{\text{True Value}}\times 100 \)

A low error rate means high accuracy and a high error rate would mean low accuracy.

### 4. Can accuracy be more than 100?

Accuracy is a deviation of the measured value from the true value. If the measure is equal to the true value, then it is taken as 100 and any variations from the true value are less than 100

Hence, accuracy cannot be more than 100

### 5. What is a good accuracy score?

To learn a good accuracy score, a range is defined across the true value, with the true value being the midpoint of the range.

Any measurement within this range is considered as a good accuracy score.

### 6. What is the difference between accuracy and precision?

Accuracy is the closeness of one value with the actual value. Precision is the closeness of one measurement with another measurement.

### 7. What is accuracy level?

A measurement equal to the actual value is said to have a good accuracy level. Any variation is said to be of low levels of accuracy.

### 8. Why does accuracy matter?

Accuracy is important, as it helps to know if the process is as per defined instruction. Also, it helps to know if a product manufactured is as per the defined specification.