Resourceaholic: Teaching Box and Whisker Plots - Free Printable

Let's analyze the boxplot of students' heights for boys and girls, and evaluate each of the 9 statements based on the visual data.

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🔍 Understanding the Boxplot Components:

A boxplot shows:
- Minimum (left end of whisker)
- First Quartile (Q1) (left edge of box)
- Median (Q2) (line inside the box)
- Third Quartile (Q3) (right edge of box)
- Maximum (right end of whisker)

We'll use these to assess the statements.

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📊 Extracting Data from the Boxplots:

#### ✔ Boys:
- Minimum: ~135 cm
- Q1: ~158 cm
- Median: ~170 cm
- Q3: ~180 cm
- Maximum: ~185 cm

#### ✔ Girls:
- Minimum: ~130 cm
- Q1: ~155 cm
- Median: ~165 cm
- Q3: ~175 cm
- Maximum: ~180 cm

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Now, let’s go through each statement:

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1. The girls are taller on average

✘ False
The median is a measure of central tendency (average in this context).
- Girls’ median = 165 cm
- Boys’ median = 170 cm

Boys have a higher median → boys are taller on average.

✔ So, this is false.

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2. The boys are taller on average

✔ True
As above, the median height of boys (170 cm) is greater than that of girls (165 cm).
So, on average, boys are taller.

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3. The girls show less spread in height

✔ True
Spread is measured by interquartile range (IQR) = Q3 – Q1

- Girls: Q3 = 175, Q1 = 155 → IQR = 20 cm
- Boys: Q3 = 180, Q1 = 158 → IQR = 22 cm

Girls have a smaller IQR → less spread in the middle 50% of data.

Also, the range (max – min):
- Girls: 180 – 130 = 50 cm
- Boys: 185 – 135 = 50 cm → same range

But since IQR is smaller for girls, they show less variability in the central portion.

✔ So, true.

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4. The boys show less spread in height

✘ False
As shown above, boys have a larger IQR (22 vs 20), so more spread.

Even though the total range is equal, the IQR (which reflects typical spread) is larger for boys.

✘ So, false.

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5. The shortest person is a girl

✔ True
- Girls’ minimum: 130 cm
- Boys’ minimum: 135 cm

So, the shortest student is a girl.

✔ True.

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6. The tallest person is a boy

✔ True
- Girls’ maximum: 180 cm
- Boys’ maximum: ~185 cm

So, the tallest person is a boy.

✔ True.

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7. Both data sets are skewed to the left

✘ False

Skewness can be inferred from the length of whiskers and position of median.

- Left-skewed: longer whisker on the left, median closer to upper quartile.
- Right-skewed: longer whisker on the right, median closer to lower quartile.

Let’s check:

#### 👦 Boys:
- Whisker on left: from 135 to 158 → length = 23 cm
- Whisker on right: from 180 to 185 → length = 5 cm
- Median at 170, which is closer to Q3 (180) than Q1 (158)

→ Longer left whisker, but median closer to Q3 → actually suggests left skew? Wait...

Wait — if the left whisker is longer, and the median is closer to Q3, that means the data is skewed to the left?

Actually, no. Let's clarify:

> A left-skewed distribution has a long tail on the left, meaning more extreme low values.

Here, boys:
- Left whisker is longer (from 135 to 158) → long tail on the left
- But median is closer to Q3 → more data on the right side

Wait — no: if the left whisker is longer, it means the tail extends to the left, so left-skewed.

But let’s think carefully:

- If the left whisker is longer, and the median is closer to Q3, then the data is left-skewed.

Yes, because the low values are more spread out, indicating a left tail.

But look at the shape:
- Boys: Q1=158, Q2=170, Q3=180 → median is closer to Q1 than Q3?
Wait! 170 is midway between 158 and 180 → 170 - 158 = 12; 180 - 170 = 10 → slightly closer to Q3.

But the left whisker goes from 135 to 158 (23 cm), while the right only 5 cm → much longer on the left.

So yes → left-skewed.

Similarly, girls:
- Min = 130, Q1 = 155 → left whisker = 25 cm
- Q3 = 175, Max = 180 → right whisker = 5 cm
- Median = 165 → between 155 and 175 → 165 - 155 = 10; 175 - 165 = 10 → centered

But left whisker much longer → left-skewed

So both have longer left whiskers, suggesting left skew.

Wait — but left-skewed means the tail is on the left, i.e., many low outliers.

But in both cases, the whisker on the left is longer, so yes, left-skewed.

But wait — in reality, human height distributions are typically right-skewed (a few very tall people), but here the left whisker is longer, so the tail is on the left, meaning more short people, so left-skewed.

So both datasets have longer left whiskers, and medians not too far from center.

But for boys, the median is closer to Q3 (170 vs 158 and 180), so slightly right-skewed? No.

Wait — if median is closer to Q3, that means the data is more concentrated on the right, so left-skewed? No!

Actually:
- If median < mean, it's right-skewed
- But we don’t have mean

Use rule of thumb:

> If median is closer to Q1, then right-skewed (because more data on the left)
> If median is closer to Q3, then left-skewed

Wait — no! Actually:

- If median is closer to Q1, then the data is skewed right (long tail on the right)
- If median is closer to Q3, then skewed left (long tail on the left)

But in boys:
- Q1 = 158, Q2 = 170, Q3 = 180
- Distance to Q1: 12 cm
- Distance to Q3: 10 cm → closer to Q3

So median is closer to Q3 → left-skewed

And left whisker is longer → confirms left skew

Similarly, girls:
- Q1 = 155, Q2 = 165, Q3 = 175
- Distance to Q1: 10 cm
- Distance to Q3: 10 cm → symmetric

But left whisker (130 to 155) = 25 cm, right (175 to 180) = 5 cm → longer left whisker → left-skewed

So both have longer left whiskers, and median close to or slightly toward Q3 → both are left-skewed

So statement 7 says: "both data sets are skewed to the left" → ✔ True

Wait — but is that accurate?

In real-world terms, heights usually aren't skewed to the left — but here, the minimum is lower, and there's a long tail to the left, so yes, the distribution is left-skewed.

But actually, left-skewed means most values are high, with a few low outliers.

That’s what we see: most boys and girls are around 160–180, but some are very short (down to 130–135), so few low values → tail on the left → left-skewed.

✔ So yes, both are left-skewed.

So 7 is TRUE.

But let's double-check.

Actually, left-skewed = negative skew = tail on the left, meaning few small values.

Yes — that’s what we have.

So 7 is TRUE.

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8. Half the boys are over 172 cm tall

✘ False

- Median of boys = 170 cm
- So 50% are below 170, 50% above

But 172 > 170, so less than 50% are over 172 cm.

Since median is 170, only those above 170 are over 170 — but 172 is higher.

We need to see if Q3 is 172? No — Q3 = 180 → 75% are under 180.

But how many are above 172?

We know:
- Median = 170 → 50% above 170
- Q3 = 180 → 75% below 180

But 172 is between 170 and 180 → so between 50% and 75% are below 172 → so between 25% and 50% are above 172.

So less than half are over 172.

Thus, not half.

So 8 is FALSE.

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9. Half the girls are under 165 cm tall

✔ True

- Median of girls = 165 cm
- By definition, 50% are below the median, 50% above

So half the girls are under 165 cm.

✔ True.

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✔ Final Answers:

| Statement | True/False | Reason |
|---------|------------|--------|
| 1. The girls are taller on average | ✘ False | Boys' median (170) > girls' (165) |
| 2. The boys are taller on average | ✔ True | Median of boys is higher |
| 3. The girls show less spread in height | ✔ True | Girls’ IQR = 20 cm, boys’ = 22 cm |
| 4. The boys show less spread in height | ✘ False | Boys have larger IQR |
| 5. The shortest person is a girl | ✔ True | Girls’ min = 130 cm, boys’ = 135 cm |
| 6. The tallest person is a boy | ✔ True | Boys’ max ≈ 185 cm, girls’ = 180 cm |
| 7. Both data sets are skewed to the left | ✔ True | Longer left whiskers, medians closer to Q3 |
| 8. Half the boys are over 172 cm tall | ✘ False | Median is 170, so only ~50% above 170; 172 is higher |
| 9. Half the girls are under 165 cm tall | ✔ True | Median = 165 cm |

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✔ Summary of True Statements:

2, 3, 5, 6, 7, 9 are true.
1, 4, 8 are false.

Let me know if you'd like a visual explanation!