Measurement and Data
Texas 6th Grade Math · Learn by Concept
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Texas 6th Grade Math › Measurement and Data
Which ordered pair is 5 units from the origin using straight-line distance (Euclidean: $\sqrt{x^2+y^2}$)?
$(3, 4)$
$(-4, -2)$
$(4, 4)$
$(2, -4.5)$
Explanation
Compute $\sqrt{x^2+y^2}$: For $(3,4)$, $\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$. For $(-4,-2)$, $\sqrt{(-4)^2+(-2)^2}=\sqrt{16+4}=\sqrt{20}\neq5$. For $(4,4)$, $\sqrt{16+16}=\sqrt{32}\neq5$. For $(2,-4.5)$, $\sqrt{4+20.25}=\sqrt{24.25}\neq5$.
A data set has these five-number summary values: min=12, Q1=18, median=24, Q3=31, max=40. What is the IQR (Q3 − Q1)?
13
11
19
28
Explanation
Compute the interquartile range: $IQR = Q_3 - Q_1 = 31 - 18 = 13$. The IQR of 13 shows the spread of the middle half of the data.
Favorite recess activities survey: Tag: 9, Four Square: 12, Jump Rope: 6, Reading: 3. What is the mode (most frequent category)?
Tag
Four Square
Jump Rope
Reading
Explanation
The mode is the category with the largest count. Four Square has 12, which is the largest, so the mode is Four Square.
Scenario A: A lab thermostat keeps the room at exactly 22.0 degrees all day with no fluctuation. Scenario B: The number of books students read this month. Scenario C: Identical coins are minted to exactly 5.00 g with no tolerance. Scenario D: The time it takes students to run one lap around the track.
Which scenarios show data with variability?
A and C
A and D
B and C
B and D
Explanation
B and D vary because different students read different amounts and run at different speeds. A and C are fixed by design, so they would not vary.
Lunch choices: Pizza: 8, Sandwich: 6, Salad: 4, Pasta: 2 (20 students). Which relative frequency table (percents) matches the data?
Pizza: 40%, Sandwich: 30%, Salad: 20%, Pasta: 10%
Pizza: 8%, Sandwich: 6%, Salad: 4%, Pasta: 2%
Pizza: 45%, Sandwich: 25%, Salad: 20%, Pasta: 10%
Pizza: 40%, Sandwich: 35%, Salad: 15%, Pasta: 10%
Explanation
Total = 20. Compute each: Pizza $= \frac{8}{20} \times 100% = 40%$, Sandwich $= \frac{6}{20} \times 100% = 30%$, Salad $= \frac{4}{20} \times 100% = 20%$, Pasta $= \frac{2}{20} \times 100% = 10%$.
Scenario A: A dispenser fills each cup with exactly 250 milliliters every time, with no error. Scenario B: Scores students earn on an open-response quiz. Scenario C: The number of minutes different buses take to arrive each morning. Scenario D: Heights of plants grown from seeds in the same potting soil.
Which scenario would most likely yield no variability?
A
B
C
D
Explanation
A is controlled to be exactly the same each time, so there is no variability. B, C, and D reflect natural differences and conditions, so they vary.
Ordered data (least to greatest): 3, 5, 5, 6, 7, 8, 9, 10, 12, 12, 14, 20. Which statement best compares the range and IQR for this set?
The range and IQR are equal.
The range is greater than the IQR.
The IQR is greater than the range.
The IQR is 17 and the range is 6.5.
Explanation
Range: $20 - 3 = 17$. For quartiles (12 values): $Q_1$ is the median of 3,5,5,6,7,8 → average of 5 and 6 is 5.5. $Q_3$ is the median of 9,10,12,12,14,20 → average of 12 and 12 is 12. So $IQR = Q_3 - Q_1 = 12 - 5.5 = 6.5$. Since 17 > 6.5, the range is greater than the IQR. This suggests the full spread is wider than the middle spread, possibly due to an outlier.
Scenario A: Each sealed bag contains exactly 20 marbles counted by a perfect machine with no mistakes. Scenario B: Daily high temperatures in a city during one week. Scenario C: The number of correct answers students get on a 2-minute fact test. Scenario D: Every pencil is cut to exactly 18.0 cm by a precise cutter with no tolerance.
Which scenarios would most likely yield no variability?
A and B
A and D
B and C
C and D
Explanation
A and D are fixed by design to be identical each time, so they show no variability. B and C change due to weather and differing student performance.
Ordered data (least to greatest): 12, 13, 15, 17, 18, 21, 22, 24, 30. What is the IQR (Q3 − Q1)?
9
7
8
12
Explanation
There are 9 values. The median is the 5th value (18). Lower half: 12, 13, 15, 17 → $Q_1$ is the average of 13 and 15, which is 14. Upper half: 21, 22, 24, 30 → $Q_3$ is the average of 22 and 24, which is 23. So $IQR = Q_3 - Q_1 = 23 - 14 = 9$. The middle half of the data varies by 9.