Question

Three apprentice tailors \((\mathrm{X}, \mathrm{Y},\) and \(\mathrm{Z})\) are assigned the task of measuring the seam of a pair of trousers. Each one makes three measurements. The results in inches are \(\mathrm{X}(31.5,31.6,31.4) ; \mathrm{Y}(32.8,32.3,32.7) ; \mathrm{Z}(31.9,\) 32.2,32.1 ). The true length is 32.0 in. Comment on the precision and the accuracy of each tailor's measurements.

Short answer

X is most precise; Z is most accurate.

Step-by-step solution

Understanding the Problem

We have three tailors: X, Y, and Z. Each has taken three measurements of the length of a pair of trousers. We need to analyze the precision and accuracy of these measurements. Precision refers to how close the measurements are to each other, and accuracy refers to how close they are to the true length (32.0 inches).

Calculating Average Measurement of Each Tailor

Calculate the mean of each tailor's measurements to see how their average compares to the true length. For tailor X: \( \text{Mean of X} = \frac{31.5 + 31.6 + 31.4}{3} = 31.5 \, \text{inches} \) For tailor Y: \( \text{Mean of Y} = \frac{32.8 + 32.3 + 32.7}{3} = 32.6 \, \text{inches} \) For tailor Z: \( \text{Mean of Z} = \frac{31.9 + 32.2 + 32.1}{3} = 32.06 \, \text{inches} \)

Analyzing Precision Using Range or Standard Deviation

Evaluate the precision by calculating the range or standard deviation (SD) of the measurements for each tailor. A smaller range or lower SD indicates higher precision. For X: Range = Max - Min = 31.6 - 31.4 = 0.2 For Y: Range = Max - Min = 32.8 - 32.3 = 0.5 For Z: Range = Max - Min = 32.2 - 31.9 = 0.3

Commenting on Precision

Tailor X's measurements have the smallest range, indicating the highest precision. Z's measurements show moderate precision, while Y's measurements have the largest range, implying lower precision.

Commenting on Accuracy

Comparing each tailor's mean measurement to the true length (32.0 inches) allows for evaluation of accuracy. X's mean (31.5 in) is less accurate as it deviates by 0.5 inches. Y's mean (32.6 in) deviates by 0.6 inches, which is less accurate. Z's mean (32.06 in) is closest to the true length, making Z's measurements the most accurate.

Key concepts

Precision

Precision refers to how closely grouped a set of measurements are to one another. It's important to differentiate this from accuracy, which is about how close a measurement is to the true value. Measuring precision involves looking at the consistency of measurements. In the exercise, each tailor took three measurements of a seam. To evaluate precision, you can look at the range or the standard deviation of their measurements.

• For Tailor X, the range was 0.2 inches.
• For Tailor Y, the range was 0.5 inches.
• For Tailor Z, the range was 0.3 inches.

A smaller range or lower standard deviation value indicates higher precision. Tailor X had the smallest range, making X's measurements the most precise.

Range

The range is one of the simplest ways to measure the spread between your highest and lowest value. It is calculated by subtracting the smallest number from the largest number in a dataset. By analyzing the range, you can gain insights into the consistency of the measurements taken.

• Tailor X: The range was 0.2 inches, indicating good precision.
• Tailor Y: The range was 0.5 inches, signifying less precision.
• Tailor Z: The range was 0.3 inches, reflecting moderate precision.

When evaluating measurements using the range, a smaller range implies the measurements are clustered closely together. Thus, X demonstrated the highest precision due to the smallest range among the three tailors.

Standard Deviation

Standard deviation (SD) provides a more detailed view of measurement spread than the range. It calculates how much values deviate from the mean, offering a snapshot of data consistency. A low standard deviation points to data that is closely packed around the mean, indicative of high precision.
To compute the standard deviation:

• First, calculate each measurement's deviation from the mean.
• Next, square each deviation to eliminate negative numbers.
• Then, calculate the average of these squared deviations.
• Finally, take the square root of that average. This is your standard deviation.

By comparing standard deviations between different datasets, you not only see which is more precise but also understand how spread out measurements are around their mean.

Mean Measurement

The mean measurement, or average, represents the central value of a set of numbers. This can provide insight into the accuracy of a series of measurements by comparing it to the known true value. To calculate the mean, simply sum all measurements and divide by the count of measurements.
For example:

• Taylor X had measurements mean of 31.5 inches, 0.5 inches below the true value of 32.0 inches.
• Tailor Y's mean was 32.6 inches, a deviation of 0.6 inches above the true length.
• Taylor Z achieved a mean of 32.06 inches, only 0.06 inches above the true value.

The closer the mean is to the true value, the more accurate the set of measurements is. In the exercise, Tailor Z's measurements were the most accurate as their mean was closest to 32.0 inches.