Dot plots, histograms and box plots

1. "More than 3" means 4, 5, or 6 books. The column above 3 is not included.
2. Count the dots in each of those columns:
   • 4 books: 3 dots
   • 5 books: 2 dots
   • 6 books: 1 dot
3. Add: 3 + 2 + 1 = 6

6 students read more than 3 books.

Check: Count the other side: 0, 1, 2, or 3 books gives 2 + 3 + 5 + 4 = 14 students, and 14 + 6 = 20 ✓ — that's every student.

Tip: "More than" and "fewer than" leave out the boundary number; "at least" and "at most" include it. Each dot is one data value — the number under the column is the value, and the number of dots is how many times it occurs.

1. A box plot is built from five numbers: minimum, first quartile (Q_1), median, third quartile (Q_3), and maximum.
2. Read them from the graph:
   • left whisker end (minimum): 52
   • left edge of box (Q_1): 70
   • line inside the box (median): 76
   • right edge of box (Q_3): 84
   • right whisker end (maximum): 96
3. The median is the line inside the box: 76.

Check: The median line sits between 75 and 80, a little closer to 75 — that's 76 ✓

Tip: The median line is usually not in the center of the box, and it is not halfway between the minimum and maximum. Always read where the line actually is.

1. "At least 4" means 4 or more, so the columns for 4, 5, and 6 all count.
2. Count the dots in each of those columns:
   • 4 times: 2 dots
   • 5 times: 1 dot
   • 6 times: 1 dot
3. Add: 2 + 1 + 1 = 4.

4 adults exercised at least 4 times.

Check: The rest exercised 0 to 3 times: 2 + 3 + 4 + 3 = 12, and 12 + 4 = 16 ✓, the total number of adults.

Tip: "At least" includes the number itself; "more than" does not. Underline these words before you count, because the GED often puts an answer for each reading.

1. "Fewer than 2" means 0 or 1 time. The 2 column is not included.
2. Count those dots: 0 times has 2, 1 time has 3. Total: 2 + 3 = 5 adults.
3. The whole group is 16 adults.
4. Fraction: \frac{5}{16}

\tfrac{5}{16} of the adults exercised fewer than 2 times.

Check: The other adults: 16 - 5 = 11 exercised 2 or more times. \tfrac{5}{16} + \tfrac{11}{16} = 1 ✓. As a decimal, 5 \div 16 = 0.3125, about 31%.

Tip: A fraction of a group always has the whole group (all the dots) on the bottom. Do not put "the rest" on the bottom.

1. A box plot shows five numbers: minimum 5, first quartile Q_1 = 12, median 18, third quartile Q_3 = 30, maximum 52.
2. The median splits the data in half: about 50% of values are below it and 50% above it.
3. The median is the line inside the box, at 18.

Half of the patients waited less than 18 minutes.

Check: The box from 12 to 30 holds the middle half. The line inside it, 18, marks the center of the data, with 25% between 12 and 18 and 25% between 18 and 30 ✓.

Tip: Box plot words to know: "half" or "50%" points to the median; "a quarter" or "25%" points to Q_1 or Q_3; "middle half" is the whole box.

1. The interquartile range is the spread of the middle half of the data: \text{IQR} = Q_3 - Q_1
2. Read the box edges: Q_1 = 70 (left edge), Q_3 = 84 (right edge).
3. Subtract: 84 - 70 = 14

The IQR is 14 points.

Check: The box covers from 70 to 84. Counting by 5s along the axis: 70 → 75 → 80 is 10, and 80 → 84 is 4 more, so 10 + 4 = 14 ✓

Tip: Range uses the whiskers (max − min); IQR uses the box (Q3 − Q1). The IQR is not affected by one extreme score, which is why statisticians like it.

1. 40 minutes or more means the intervals 40–under 50 and 50–under 60.
2. Bar heights: 5 and 3. Add: 5 + 3 = 8 \text{ workers}
3. Divide by the total of 50 workers and change to a percent: \frac{8}{50} = 0.16 = 16\%

16% of the workers commute 40 minutes or more.

Check: 16\% of 50 is 0.16 \times 50 = 8 ✓

Tip: The total number of data values in a histogram is the sum of all bar heights: 5 + 17 + 12 + 8 + 5 + 3 = 50. Use that total as the whole when you find a percent.

1. The five numbers split the data into four parts, each holding about 25% of the patients:
   • 5 to 12 (left whisker): 25%
   • 12 to 18 (left part of box): 25%
   • 18 to 30 (right part of box): 25%
   • 30 to 52 (right whisker): 25%
2. From 12 (Q_1) to 52 (maximum) covers three of these parts.
3. 3 \times 25\% = 75\%.

About 75% of the patients waited between 12 and 52 minutes.

Check: The only part left out is 5 to 12, which is 25%. 100\% - 25\% = 75\% ✓.

Tip: On a box plot, each section holds about the same number of values, no matter how long or short it looks.

1. "At least 40" means 40 or older. The bars that start at 40 or higher are 40–50, 50–60, and 60–70.
2. Read their heights:
   • 40 to under 50: 11
   • 50 to under 60: 9
   • 60 to under 70: 6
3. Add: 11 + 9 + 6 = 26.

26 people were at least 40.

Check: People under 40: 6 + 16 + 12 = 34. 34 + 26 = 60 ✓, the whole audience.

Tip: Because each interval includes its left edge, a 40-year-old is in the 40–50 bar. Read the interval rule in the stem every time.

Five-number summaries:

• Fertilizer: 14, 20, 24, 27, 33
• None: 10, 15, 19, 22, 30

1. Medians: 24 - 19 = 5. The fertilizer median is 5 cm greater. True.
2. IQRs: fertilizer 27 - 20 = 7; none 22 - 15 = 7. They are equal, so "greater IQR" is false.
3. Ranges: fertilizer 33 - 14 = 19; none 30 - 10 = 20. The no-fertilizer range is larger, not smaller.
4. Overlap: The no-fertilizer maximum (30) is taller than the fertilizer minimum (14), so not every fertilized plant was taller.

The true statement is the first one: the median is 5 cm greater with fertilizer.

Check: 19 + 5 = 24 ✓.

Tip: Compare centers with medians and spreads with IQR or range. A plot that sits farther right does not automatically have more spread.

1. There are 50 values, so the median is the average of the 25th and 26th values (in order from shortest to longest).
2. Add the bar heights from the left (a running total):
   • 0–under 10: 5 → values 1–5
   • 10–under 20: 5 + 17 = 22 → values 6–22
   • 20–under 30: 22 + 12 = 34 → values 23–34
3. The 25th and 26th values are both in values 23–34, so they are in the 20 to under 30 minutes interval.

The median is in the 20–30 minute interval.

Check: Fewer than half the workers (22) are below 20 minutes, and more than half (34) are below 30 minutes, so the middle worker is between 20 and 30 ✓

Tip: A histogram can't give you the exact median, but it can tell you which interval holds it. The tallest bar shows the most common interval (the modal interval), which is a different thing.

1. There are 60 people (an even number), so the median is the average of the 30th and 31st ages when they are listed in order.
2. Keep a running total from the youngest bar:
   • 10 to under 20: 6 (people 1–6)
   • 20 to under 30: 6 + 16 = 22 (people 7–22)
   • 30 to under 40: 22 + 12 = 34 (people 23–34)
3. The 30th and 31st people are both between 23 and 34, so they are in the 30 to under 40 interval.

The median age is in 30 to under 40.

Check: 22 people are younger than 30 (fewer than half) and 34 are younger than 40 (more than half), so the middle must be in the 30s ✓.

Tip: The tallest bar shows the most common interval, not the median. Running totals find the middle every time.

1. "Less than 10 hours" covers the intervals 6 to under 8 and 8 to under 10.
2. Add those bars: 4 + 12 = 16 phones.
3. Divide by the whole, 40 phones: \frac{16}{40} = 0.40
4. As a percent: 0.40 \times 100 = 40\%.

40% of the phones lasted less than 10 hours.

Check: The phones lasting 10 hours or more: 9 + 11 + 4 = 24, and \frac{24}{40} = 60\%. 40\% + 60\% = 100\% ✓.

Tip: Always check whether the question wants one bar or several. "Less than 10" includes everything to the left of 10.

1. There are 40 phones, so the median is the average of the 20th and 21st values in order.
2. Running totals from the left:
   • 6 to under 8: 4 (phones 1–4)
   • 8 to under 10: 4 + 12 = 16 (phones 5–16)
   • 10 to under 12: 16 + 9 = 25 (phones 17–25)
3. Positions 20 and 21 are in the 17–25 group, the 10 to under 12 interval.

The median battery life is in 10 to under 12 hours.

Check: 16 phones are below 10 hours (less than half of 40) and 25 are below 12 hours (more than half) ✓.

Tip: The tallest bar here (8 to under 10) is a trap: it is the modal interval. The median depends on positions, not on the height of one bar.

1. The data are already in order. Minimum = 4, maximum = 11.
2. Median: 11 values, middle is position 6: 7.
3. Q_1: median of the lower five values 4, 5, \mathbf{6}, 6, 7 → 6.
4. Q_3: median of the upper five values 8, 8, \mathbf{8}, 9, 11 → 8.
5. Five-number summary: 4, 6, 7, 8, 11.
6. Compare: Plot A has median 8, Plot B has whiskers at 5 and 9, Plot D has median 7.5. Only Plot C shows 4, 6, 7, 8, 11.

The correct plot is Plot C.

Check: In Plot C, the whiskers reach the true smallest and largest values (4 and 11), and the median line sits at 7, with 5 values on each side ✓.

Tip: To match a box plot, check all five numbers. Wrong choices often get four right and one wrong.

Before (13 buses):

1. Total minutes: 0(4) + 1(2) + 2(3) + 3(1) + 4(2) + 8(1) = 0 + 2 + 6 + 3 + 8 + 8 = 27.
2. Mean: 27 \div 13 \approx 2.08. Median: position 7 = 2.

After removing 8 (12 buses): 3. Total: 27 - 8 = 19. Mean: 19 \div 12 \approx 1.58. 4. Ordered: 0, 0, 0, 0, 1, \mathbf{1}, \mathbf{2}, 2, 2, 3, 4, 4. Median = \dfrac{1 + 2}{2} = 1.5.

Both the mean and the median decrease (mean 2.08 \to 1.58, median 2 \to 1.5).

Check: 1.58 \times 12 \approx 19 ✓, and 6 values lie on each side of 1.5 ✓.

Tip: Removing a value changes the count, so the median can shift too, especially with a small data set. Do not assume the median never moves.

With 7 values in order, the median is the 4th value, Q_1 is the median of values 1–3 (the 2nd value), and Q_3 is the median of values 5–7 (the 6th value).

1. 2, 3, 4, 6, 9, 10, 12: median 6, but Q_1 = 3 and Q_3 = 10. No.
2. 2, 4, 5, 6, 8, 9, 12: min 2, Q_1 = 4, median 6, Q_3 = 9, max 12. Yes.
3. 2, 4, 6, 7, 8, 9, 12: median = 7. No.
4. 2, 4, 5, 6, 9, 11, 12: Q_3 = 11. No.

The list 2, 4, 5, 6, 8, 9, 12 matches.

Check: In that list, 3 values are below the median and 3 above. Lower half 2, 4, 5 has middle 4; upper half 8, 9, 12 has middle 9 ✓.

Tip: Working backwards from a box plot, test each answer choice for all five numbers. The minimum and maximum are usually the same in every choice, so focus on the median and quartiles.

1. Read the five numbers: minimum $85, Q_1 = $110, median $125, Q_3 = $150, maximum $205.
2. The top 25% of the data lies in the right whisker, from Q_3 to the maximum.
3. That section begins at Q_3 = 150.

The highest 25% of the bills were all at least $150.

Check: 75% of the bills are at or below Q_3 and 25% are at or above it. With 48 bills, that is about 0.25 \times 48 = 12 bills from $150 to $205 ✓.

Tip: "Top 25%" starts at Q_3; "bottom 25%" ends at Q_1; "top half" starts at the median.

1. New mean. The old total was 2.65 \times 20 = 53 books. Add the new student: 53 + 13 = 66 \text{ books}, \qquad \frac{66}{21} \approx 3.14
2. New median. Now there are 21 values; the middle is position \dfrac{21 + 1}{2} = 11.
   • Running total: 0 books → 2, 1 book → 5, 2 books → 10, 3 books → 14.
   • The 11th value is in the 3-book column, so the new median is 3. (13 goes at the far right end, which shifts the middle position one step to the right.)
3. So the mean rises from 2.65 to about 3.14, and the median rises from 2.5 to 3.

The correct statement: the mean increases to about 3.14 and the median increases to 3.

Check: 3.14 \times 21 \approx 66 ✓, and in the new list 10 values (0, 0, 1, 1, 1, 2, 2, 2, 2, 2) come before the 11th value, a 3, and 10 values come after it ✓

Tip: An outlier pulls the mean toward it a lot. The median moves at most one position, no matter how extreme the new value is. That is why the median is preferred for data with outliers.

Mean

1. Original total: 0(4) + 1(2) + 2(3) + 3(1) + 4(2) + 8(1) = 27; original mean = 27 \div 13 \approx 2.08.
2. New total: 27 + 20 = 47; new mean = 47 \div 14 \approx 3.36.
3. Change: 3.36 - 2.08 \approx 1.28, an increase of more than 1 minute.

Median 4. New ordered list (14 values): 0, 0, 0, 0, 1, 1, \mathbf{2}, \mathbf{2}, 2, 3, 4, 4, 8, 20. 5. Median = \dfrac{2 + 2}{2} = 2, the same as before.

The mean increases by more than 1 minute, and the median stays at 2.

Check: 3.36 \times 14 \approx 47 ✓. Seven values are at or below 2 and seven at or above 2 ✓.

Tip: An extreme value pulls the mean toward it but barely moves the median. That is why the median is often the better "typical" value when there are outliers.