Question

derriyah hudson
worksheet 1: representing data (s.id.1)
part a: raw data
the following data set shows the number of minutes students spent studying for a math
test:
45, 30, 60, 90, 45, 75, 30, 60, 45, 120, 90, 60

1. create a dot plot to represent the data.
2. create a histogram using intervals of 30 minutes.
3. create a box plot.

part b: analysis

1. which representation best shows clusters? explain.
2. which representation best shows spread? explain.

Explanation:

Step1: Organize data by frequency

First, sort and count occurrences of each value:
30: 2, 45: 3, 60: 3, 75:1, 90:2, 120:1

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1. Dot Plot Creation

Step1: Label axis with values

Mark x-axis with the unique study times: 30, 45, 60, 75, 90, 120

Step2: Plot dots for each occurrence

• 30: ● ●
• 45: ● ● ●
• 60: ● ● ●
• 75: ●
• 90: ● ●
• 120: ●

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2. Histogram Creation

Step1: Define 30-minute intervals

Intervals: 0-30, 31-60, 61-90, 91-120

Step2: Count data per interval

• 0-30: 2 (values: 30,30)
• 31-60: 8 (values: 45,45,45,60,60,60)
• 61-90: 3 (values:75,90,90)
• 91-120:1 (value:120)

Step3: Draw histogram bars

• x-axis: intervals; y-axis: frequency
• Bar 0-30: height=2
• Bar 31-60: height=8
• Bar 61-90: height=3
• Bar 91-120: height=1

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3. Box Plot Creation

Step1: Sort the full data set

Sorted data: $30, 30, 45, 45, 45, 60, 60, 60, 75, 90, 90, 120$

Step2: Calculate 5-number summary

• Minimum: $30$
• Q1 (25th percentile): $\frac{45+45}{2}=45$
• Median (50th percentile): $\frac{60+60}{2}=60$
• Q3 (75th percentile): $\frac{75+90}{2}=82.5$
• Maximum: $120$

Step3: Draw box plot

• Draw a number line, mark the 5 summary values. Draw a box from Q1 to Q3, a line at the median, and whiskers from min to Q1, Q3 to max.

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4. Cluster Representation Analysis

Step1: Identify best plot for clusters

Dot plots show individual data points grouped at specific values, making clusters visually clear.

Step2: Explain reasoning

Clusters are groups of similar values; the dot plot directly shows dense groupings at 45 and 60, which is easier to see than in a histogram or box plot.

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5. Spread Representation Analysis

Step1: Identify best plot for spread

Box plots explicitly display the 5-number summary, which quantifies the full range and interquartile spread of the data.

Step2: Explain reasoning

The box plot's whiskers show the total spread (min to max), and the box shows the spread of the middle 50% of data, which is designed to highlight data dispersion more directly than the other plots.

Answer:

1. Dot Plot:

    30: ● ●
    45: ● ● ●
    60: ● ● ●
    75: ●
    90: ● ●
    120: ●

1. Histogram:

Interval (Minutes)	Frequency
31-60	8
61-90	3
91-120	1

(Bars are drawn with heights matching the frequencies above)

1. Box Plot:

• Number line marked with: 30 (min), 45 (Q1), 60 (median), 82.5 (Q3), 120 (max)
• Box spanning 45 to 82.5, median line at 60, whiskers from 30 to 45 and 82.5 to 120

1. The dot plot best shows clusters. It displays individual data points, so dense groupings (clusters) of values like 45 and 60 are immediately visible as groups of dots.
2. The box plot best shows spread. It directly displays the minimum, maximum, and quartiles, which clearly quantify the total range of the data and the spread of the middle 50% of values.