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Question 1

How long is a minute?

Two groups of children were asked to estimate the length of 1 minute. They start by closing their eyes and then open them when they think 1 minute has passed. Of course, they are not supposed to count while their eyes are closed. The dot plots below show after how many seconds the children opened their eyes.

Discuss how well both the groups fared at this activity. Describe and compare the variability in data and their central tendency.

Solution

Understand the Question

The actual length of $1\text{ minute}$ is $60\text{ seconds}$ .
Central Tendency (Mean and Median): Indicates how close the average and middle estimates are to $60\text{ seconds}$ .
Variability (Range and Spread): Indicates how scattered or consistent the estimates are: $\text{Range} = \text{Maximum Value} - \text{Minimum Value}$

Step 1 · Analyze Group A's Estimates

Diagram 1

From the dot plot for Group A:

$\text{Mean} = 58.21\text{ seconds}$
$\text{Median} = 60\text{ seconds}$
$\text{Minimum} = 42\text{ seconds}$
$\text{Maximum} = 69\text{ seconds}$

\begin{aligned} \text{Range} &= 69 - 42 \\ &= 27\text{ seconds} \end{aligned}

Step 2 · Analyze Group B's Estimates

Diagram 2

From the dot plot for Group B:

$\text{Mean} = 59.28\text{ seconds}$
$\text{Median} = 59.5\text{ seconds}$
$\text{Minimum} = 45\text{ seconds}$
$\text{Maximum} = 71\text{ seconds}$

\begin{aligned} \text{Range} &= 71 - 45 \\ &= 26\text{ seconds} \end{aligned}

Step 3 · Compare Central Tendency and Variability

1. Central Tendency:

Group A has a median of exactly $60\text{ seconds}$ , while Group B has a mean of $59.28\text{ seconds}$ and a median of $59.5\text{ seconds}$ .
Both groups showed high accuracy as their central values are extremely close to $60\text{ seconds}$ .

2. Variability:

Group A has a range of $27\text{ seconds}$ , and Group B has a range of $26\text{ seconds}$ .
Although both ranges are very similar, Group B's estimates are more tightly clustered around $60\text{ seconds}$ , indicating slightly greater consistency.

Answer

Both groups estimated $1\text{ minute}$ accurately on average. Group A achieved an exact median of $60\text{ s}$ , while Group B had a mean of $59.28\text{ s}$ and median of $59.5\text{ s}$ . In terms of variability, both had similar ranges ( $27\text{ s}$ for Group A and $26\text{ s}$ for Group B), but Group B was slightly more consistent due to tighter clustering around $60\text{ s}$ .

Common Mistakes

Overlooking Data Clustering: Assuming two data sets have the same spread merely because their ranges are nearly equal ( $27\text{ s}$ vs. $26\text{ s}$ ), without checking how tightly points cluster around the center.
Single Measure Bias: Evaluating performance using only the mean or only the median instead of comparing both measures of central tendency.

Question 1

How long is a minute?

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How long is a minute?