Frequency Tables and Histograms Worksheet for 5th Grade | Lesson ... - Free Printable

Let’s solve each question step by step using the histogram titled “United States Presidents, Age at Inauguration.”

The x-axis shows age groups (bins):
- 40–44
- 45–49
- 50–54
- 55–59
- 60–64
- 65–69

The y-axis shows frequency (number of presidents).

From the bar heights (reading carefully):

- 40–44: height = 2 → 2 presidents
- 45–49: height = 7 → 7 presidents
- 50–54: height = 8 → 8 presidents
- 55–59: height = 13 → 13 presidents
- 60–64: height = 8 → 8 presidents
- 65–69: height = 3 → 3 presidents

Total presidents shown = 2 + 7 + 8 + 13 + 8 + 3 = 41

(Note: As of 2026, there have been 46 U.S. presidents, but this histogram may be from an older textbook or exclude some — we’ll use only what’s shown.)

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Question 1: How many presidents were between the ages of 55 and 59 at the time of their inauguration?

Look at the bar for 55–59. Its height is 13.

✔ Answer: 13

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Question 2: Identify a summarizing data title. The data represents the number of U.S. Presidents inaugurated at different age ranges. What does the bin size tell you about how old they were? Do you agree with it? Why or why not?

Wait — let’s read the question again:

> “Identify a summarizing data title. The data represents... [already given]. What does the bin size tell you about how old they were? Do you agree with it? Why or why not?”

Actually, looking back at the image text:

> “2. Identify a summarizing data title. The data represents the number of U.S. Presidents inaugurated at different age ranges. What does the bin size tell you about how old they were? Do you agree with it? Why or why not?”

This seems to ask two things:

a) Suggest a better/summarizing title? But the current title is already clear: “United States Presidents, Age at Inauguration”

But maybe it wants us to interpret the *bin size*.

Bin size here is 5 years (each group spans 5 years: 40–44, 45–49, etc.)

What does that tell us? It means we’re grouping presidents into 5-year chunks to see patterns — like most were inaugurated in their 50s.

Do I agree with using 5-year bins? Yes — because:

- Too small (like 1-year bins) would make too many bars and be messy.
- Too large (like 10-year bins) might hide important details (e.g., difference between early 50s vs late 50s).
- 5-year bins are standard for age data and show trends clearly.

Also, note: The first bin starts at 40 — which makes sense since no president was inaugurated younger than 40 (youngest was Theodore Roosevelt at 42). Last bin ends at 69 — oldest was Ronald Reagan at 69 when inaugurated second term.

So yes, bin size of 5 is appropriate.

✔ Summary answer:

A good summarizing title could be: “Age Distribution of U.S. Presidents at Inauguration”

The bin size of 5 years tells us that presidents’ ages are grouped every 5 years to show general trends without getting too detailed. I agree with this choice because it balances clarity and detail — showing where most presidents fall (55–59) while keeping the graph easy to read.

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Question 3: If the histogram were redrawn with 8-year intervals instead of 5, how many presidents would be included in the first interval?

First interval if we start at 40 and use 8-year bins:

→ First bin: 40–47 (since 40 + 8 = 48, so up to but not including 48? Or inclusive?)

In histograms, bins are usually left-inclusive, right-exclusive — but sometimes labeled inclusively. Let’s assume based on original labeling (40–44 includes both endpoints), so likely inclusive.

But 8-year interval starting at 40: 40 to 47 inclusive? That’s 8 years: 40,41,42,43,44,45,46,47 → 8 values.

Original bins:

- 40–44: 2 presidents
- 45–49: 7 presidents → but only part of this falls in 40–47

If new bin is 40–47, then:

- All of 40–44 → 2
- Part of 45–49 → specifically ages 45,46,47 → but we don’t know exact distribution within 45–49!

Problem: We can’t split the 7 presidents in 45–49 unless we assume uniform distribution — which isn’t stated.

But wait — perhaps the question expects us to combine full bins that fit entirely or mostly?

Alternative interpretation: Maybe “first interval” means 40–47, and since 45–49 overlaps, we take all of 40–44 and none of 45–49? That doesn’t make sense.

Or perhaps re-bin as:

New bins of 8 years starting at 40:

Bin 1: 40–47
Bin 2: 48–55
Bin 3: 56–63
Bin 4: 64–71

Now map original data:

Original:

- 40–44 → all in 40–47 → 2
- 45–49 → 45,46,47 go to bin1; 48,49 go to bin2 → but we don’t know how many of the 7 are in 45–47 vs 48–49

Same problem.

Unless... perhaps the question assumes we just add entire original bins that fall completely or partially? But that’s inaccurate.

Wait — look at total presidents: 41

Perhaps the intended approach is to realize that with 8-year bins starting at 40:

First bin: 40–47

Which original bins contribute?

- 40–44: fully inside → 2
- 45–49: partially inside → but since we don’t have sub-data, maybe we can’t determine exactly?

But that can’t be — this is a school problem. Likely, they expect us to assume that the 7 in 45–49 are evenly distributed? Or perhaps ignore partial overlap?

Another idea: Maybe “first interval” means 40–47, and since 45–49 starts at 45, which is >47? No, 45<47.

I think there's ambiguity, but let’s try to estimate reasonably.

Assume the 7 presidents in 45–49 are spread evenly across those 5 years: 45,46,47,48,49 → 7 presidents over 5 years ≈ 1.4 per year.

Then ages 45,46,47 → 3 years → 3 × 1.4 = 4.2 → approx 4 presidents

Plus 2 from 40–44 → total ≈ 6

But we can't have fraction.

Alternatively, maybe the problem intends for us to include only full original bins that lie entirely within the new first bin? But 45–49 doesn't lie entirely in 40–47.

Wait — perhaps the new binning starts differently? Or maybe "first interval" means 40–47, and we take all presidents aged ≤47.

From original:

- 40–44: 2 (all ≤44 <47)
- 45–49: we need those ≤47 → i.e., 45,46,47

But again, no breakdown.

Perhaps in context of the course, they teach to combine bins conservatively? Or maybe I misread.

Let me check total again.

Another thought: Maybe the first 8-year interval is 40–47, and since the next original bin is 45–49, which overlaps, but perhaps for simplicity, they want us to take only the first full bin? That would be 2 — but that ignores 45–47.

I recall that in some curricula, when rebinnning, if a bin overlaps, you assign based on midpoint or something — but not specified.

Wait — let's calculate how many presidents are under 48.

Ages 40 to 47 inclusive.

From histogram:

- 40–44: 2
- 45–49: 7 — but these are ages 45 to 49. So if we assume the 7 are distributed, say, 2 at 45, 2 at 46, 2 at 47, 1 at 48 — but arbitrary.

Perhaps the problem has a typo, or expects us to use only complete bins.

Let’s look at the last part of Q3: “how many presidents would be included in the first interval?”

And Q4 asks about shape change — so probably Q3 is meant to be calculable.

Another idea: Perhaps "8-year intervals" means bins like 40-47, 48-55, etc., and for the first bin 40-47, it includes:

- All of 40-44 (2)
- And the portion of 45-49 that is <=47, which is 45,46,47 — 3 out of 5 years.

If we assume uniform distribution in 45-49, then 3/5 of 7 = 4.2 — round to 4.

So 2 + 4 = 6.

Since number of presidents must be integer, and 4.2 is closer to 4, let's go with 6.

But is there a better way?

Let's list all possible ages and frequencies if we distribute evenly.

Suppose in 45-49 bin (7 presidents), assume:

Age 45: 1.4
46: 1.4
47: 1.4
48: 1.4
49: 1.4

Then for 40-47: ages 40-44: 2, plus 45,46,47: 1.4*3=4.2, total 6.2 — still not integer.

Perhaps the 7 are whole numbers: e.g., 2 at 45, 2 at 46, 2 at 47, 1 at 48 — then 45-47: 6, plus 2 from 40-44 = 8? But 48 is not in first bin.

If first bin is 40-47, then 48 is excluded.

So if 45-49 has 7 presidents, and we put 3 years in first bin (45,46,47), and 2 years in second (48,49), then if distributed proportionally, 3/5 *7 = 4.2, so approximately 4.

Thus 2 + 4 = 6.

I think 6 is the expected answer.

To confirm, let's see what happens if we do the same for other bins later — but for now, I'll go with 6.

✔ Answer for Q3: 6

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Question 4: What does the shape of the histogram tell you about the ages at which presidents are inaugurated?

Shape: Looking at the bars:

- Low at 40-44 (2)
- Rises to 45-49 (7)
- Peaks at 55-59 (13)
- Then decreases: 60-64 (8), 65-69 (3)

So it's unimodal (one peak), skewed slightly to the right? Wait — peak at 55-59, then drops, but 60-64 is 8, which is higher than 45-49's 7, so not symmetric.

Actually, from left to right: 2,7,8,13,8,3

So it rises to a peak at 55-59, then falls. The left side (younger) has lower frequencies, right side (older) also lower, but the drop after peak is steeper on the right? From 13 to 8 to 3 — yes, steeper decline on right.

But typically, if peak is not in center, and tail is longer on one side.

Here, minimum age 40, maximum 69, peak at 55-59, which is closer to center? Center of range 40-69 is 54.5, so 55-59 is very close to center.

Frequencies: left of peak: 2+7+8=17, right of peak: 8+3=11, so more on left? But peak is at 55-59, so left includes 40-54, right 60-69.

Number before peak: 40-44,45-49,50-54: 2+7+8=17
At peak: 13
After peak: 60-64,65-69: 8+3=11

So total before peak: 17, after: 11, so slightly more younger presidents, but the peak is high.

The shape is roughly bell-shaped but skewed left? No — if more mass on left, it's skewed right? Standard definition: skewness direction is where the tail is.

Tail on right: from 59 to 69, frequencies 8,3 — decreasing, but 8 is still substantial.

Compared to left: from 40 to 55, increasing from 2 to 13.

Actually, the distribution is somewhat symmetric around 55-59, but not perfectly.

Mean might be around 55.

But simply: the histogram shows that most presidents are inaugurated in their mid-to-late 50s, with fewer at younger and older ages. The shape is unimodal and approximately symmetric, though slightly skewed to the right (since the right tail goes down to 3, while left starts at 2, but the rise is gradual).

More accurately: since the peak is at 55-59, and the frequencies decrease on both sides, but the decrease on the right (to 3) is sharper than on the left (from 2 to 7 to 8), actually the left has a longer "tail" in terms of number of bins, but values are smaller.

Perhaps best to say: The shape is unimodal with a peak at 55-59 years, indicating that this is the most common age range for presidential inauguration. The distribution is roughly symmetric but with a slight right skew, meaning there are a few presidents inaugurated at older ages compared to younger ones, but overall, ages cluster around the mid-50s.

For a student, keep it simple.

✔ Answer: The histogram is highest in the middle (55-59 years) and lower on both ends, which means most presidents are inaugurated in their mid-50s. It looks like a hill, showing that very young or very old inaugurations are less common.

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Now, compiling final answers.

Note: For Q2, it's descriptive, so I'll write a concise version.

Final Answers:

1. 13
2. A good title is "Age Distribution of U.S. Presidents at Inauguration." The bin size of 5 years groups ages to show trends clearly. I agree because it avoids too much detail while highlighting that most presidents are in their 50s.
3. 6
4. The shape shows that most presidents are inaugurated between 55 and 59 years old, with fewer at younger or older ages. It peaks in the middle, like a hill, meaning mid-50s is the most common age.

But according to instructions, Final Answer section should provide only the final correct answer — and since there are multiple questions, I need to present them clearly.

Looking back at user request: "Solve the problem accurately." and "Final Answer: [Provide only the final correct answer]"

But there are four questions. Probably, I should list answers for each.

In the response format, it says "Final Answer: [Provide only the final correct answer]" — singular, but likely means for the whole task.

To be safe, I'll provide all answers in Final Answer section, numbered.

Also, for Q2 and Q4, they are explanatory, so I'll summarize key points.

Let me write:

Final Answer:
1. 13
2. Summarizing title: "Distribution of U.S. Presidential Inauguration Ages". Bin size of 5 years allows clear visualization of age clusters; agreed upon as it balances detail and readability, showing peak in 50s.
3. 6
4. The histogram is unimodal and roughly symmetric, peaking at 55-59 years, indicating most presidents are inaugurated in their mid-50s, with fewer at extremes.

But to strictly follow "only the final correct answer", and since Q2 and Q4 require explanation, perhaps in Final Answer I put numerical answers and brief phrases.

I think it's acceptable to include short answers for all.

Alternatively, since the user said "solve the problem", and problem has parts, answer all parts.

I'll proceed.

One more check on Q3: Is 6 correct?

Let me double-check.

Original bins:

40-44: 2
45-49: 7
50-54: 8
55-59: 13
60-64: 8
65-69: 3

New bins of 8 years, starting at 40:

Bin 1: 40-47
Bin 2: 48-55
Bin 3: 56-63
Bin 4: 64-71

Now, assign:

- 40-44: all in bin1 → 2
- 45-49: ages 45,46,47 in bin1; 48,49 in bin2
Assume uniform: 7 presidents over 5 years → per year 1.4
So 45,46,47: 3 years * 1.4 = 4.2 → round to 4
Thus bin1 gets 2 + 4 = 6
Bin2 gets remaining 3 from 45-49? 7 - 4 = 3, but also 50-54: all in bin2? 50-54 is within 48-55? 50 to 54 < 55, yes, if bin2 is 48-55 inclusive.

48-55 includes 48,49,50,51,52,53,54,55 — 8 years.

50-54: 8 presidents, all in bin2.

55-59: 55 is in bin2? If bin2 is 48-55, and 55 is included, then 55 is in bin2, 56-59 in bin3.

This is messy.

Perhaps define bins as [40,48), [48,56), [56,64), [64,72) — half-open intervals.

Then:

Bin1: [40,48) → ages 40 to 47.999, so 40-47 inclusive.

Original:

40-44: all in [40,48) → 2
45-49: 45,46,47 in [40,48); 48,49 in [48,56)
So if 7 presidents in 45-49, and assuming equal likelihood, probability age <48 is 3/5 (ages 45,46,47 out of 45-49), so expected number in bin1 from this bin: (3/5)*7 = 4.2

Similarly, 50-54: all >=50 >48, so in [48,56)
55-59: 55 in [48,56)?