EST I HS Math · Unit 2 · Data Analysis & Probability
Lesson 2.5 · Tables, Bar Graphs & Two-Way Tables
On the paper: A frequency table or a two-way table shows up on nearly every real paper
Big idea. A table and a bar chart are the same information in two shapes. Read every row/column total FIRST — most of these questions are really a percentage or a probability question wearing a table's clothes.
1Frequency tables: mode, median, mean 📊
🍼 Baby steps
- 1
siblings 0,1,2,3,4 → Company B: 25,40,25,15,20 (total 125)
October 2020. Read the table as (value, how many people have that value).
- 2
median position = (125+1)/2 = 63rd value
The median POSITION depends on the total count, not the number of rows in the table.
- 3
cumulative: 0→25, 1→65 → the 63rd value falls in "1" → median = 1
Add up frequencies row by row until you pass the target position.
- 4
mode = the value with the LARGEST frequency, not the largest value
Company B's biggest frequency (40) sits at value "1" → mode = 1.
Trap · two tables, one comparison
2Two-way tables: rows × columns ⚏
🍼 Baby steps
- 1
rows = age group, columns = sports activity, + a Total row and a Total column
August 2021. Every cell answers "how many people are BOTH this row AND this column."
- 2
"if an employee aged UNDER 40 is picked, P(practices sport)?" → use the UNDER-40 ROW's own total, not the grand total
220 practice sport out of 260 people who are under 40 → 220/260 = 11/13.
- 3
"given that a person works OUTSIDE their field, P(they majored in business)?" → use that COLUMN's total
August 2021. 11,753 business-outside-field, out of 23,784 total outside-field → ≈ 0.5.
🎛 Play with it · one dataset, two shapes
Slide a bar to change one value — watch the table and the mode/median/mean update together, since they're all reading the exact same numbers.
As a bar chart
As a frequency table
| Value | Frequency |
|---|---|
| 0 | 15 |
| 1 | 45 |
| 2 | 30 |
| 3 | 25 |
| 4 | 10 |
| Total | 125 |
mode = 1 · median = 2 · mean = 1.76
3A bar graph as a percentage word problem 📈
🍼 Baby steps
- 1
a bar graph shows % of a company's budget spent per sector
June 2021. Transport's BAR = a percentage of the total budget, not a dollar amount by itself.
- 2
if $200,000 IS the transport sector's %, find the total budget, then salaries' %
Read the graph for the two percentages, use the given dollar figure to find the whole, then apply salaries' %.
4Real EST questions, solved twice 📄
A survey grouped employees of two companies by their number of siblings: Company B had 25 employees with 0 siblings, 40 with 1, 25 with 2, 15 with 3, and 20 with 4. What is the median number of siblings for Company B?
- A.1
- B.2
The median of the largest single frequency's row (Company A), not Company B.
- C.3
No route to this value from the table.
- D.4
The largest value in the table, not the median.
Safe way · ~30 s
- Total = 25+40+25+15+20 = 125. Median position = 63rd.
- Cumulative: 0→25, 1→65. The 63rd value falls in "1".
⏱ 90-second way · ~12 s
- Half of 125 is 62.5 — the middle value is inside the "1" group once you pass 25.
Answer: A
A rabbit farm has small and big rabbits in white, brown, and gray. Small: 4 white, 4 brown, 6 gray. Big: 10 white, 2 brown, 4 gray. One rabbit is picked at random. What is the probability that it is white?
- A.4/14
Only counts the small-white rabbits, ignoring the big-white ones.
- B.7/15
- C.1
Not a valid probability move here.
- D.10/30
Only counts the big-white rabbits.
Safe way · ~30 s
- Total rabbits = (4+4+6)+(10+2+4) = 14+16 = 30.
- White rabbits = 4 (small) + 10 (big) = 14. P = 14/30 = 7/15.
⏱ 90-second way · ~12 s
- Add the White column (4 + 10 = 14), divide by the grand total (30).
Answer: B
City A: 25, 31, 14, 8, 2 households own 1, 2, 3, 4, 5 cars respectively. City B: 20, 19, 23, 12, 6. What is the mean number of cars owned, combining both cities?
- A.Between 1 and 2
Ignores that "2 cars" and "3 cars" are the two biggest combined groups.
- B.Between 2 and 3
- C.Between 3 and 4
Overestimates — most households own 1-3 cars, not 3-5.
- D.Between 4 and 5
Far too high for this data.
Safe way · ~40 s
- Combine: 1→45, 2→50, 3→37, 4→20, 5→8. Total = 160.
- Weighted sum = 45+100+111+80+40 = 376. Mean = 376 ÷ 160 = 2.35.
⏱ 90-second way · ~15 s
- Combine the two tables cell by cell first — never average two means directly when the group sizes differ.
Answer: B
A table of 540 employees by age (under 40 / 40 and older) and sports activity shows: under 40 — 220 practice sport, 40 don't (260 total); 40-and-older — 100 practice sport, 180 don't (280 total). If an employee AGED UNDER 40 is picked at random, what is the probability that they practice sport?
- A.2/13
Uses the wrong row entirely.
- B.11/27
Divides by the grand total (540) instead of the under-40 row's own total.
- C.11/16
Mixes the under-40 sport count with the 40-and-older row's total.
- D.11/13
Safe way · ~25 s
- "Aged under 40" restricts you to that ROW only: total 260.
- 220 of those 260 practice sport → 220/260 = 11/13.
⏱ 90-second way · ~10 s
- 220 ÷ 260 simplifies to 11/13 — the condition picks the row, not the grand total.
Answer: D
⏱ The 90-second lane
One question = 90 seconds. These are the moves that finish this topic in one line.
- 1A frequency table's median position depends on the TOTAL count — add up 1+ (total ÷ 2) rounded appropriately, then find which row that position falls in.
- 2Mode = the value with the biggest frequency, not the biggest value in the table.
- 3"Comparing two groups" questions need mode, median, AND mean computed for BOTH before you can say which one matches.
- 4In a two-way table, "given that…" or "if a _____ is picked" restricts you to ONE row or ONE column — divide by THAT total, never the grand total.
- 5A percentage bar graph gives you two %s and one real number — use the real number to find the total, then apply the other %.
Take with you
- ▸A table and a bar/pie chart show the exact same data — practice reading both, and converting between them.
- ▸Median needs the total count; mode needs the biggest FREQUENCY, not the biggest value.
- ▸A two-way table's conditional questions ("given that…") always restrict you to one row or column's own total.