Distributions: Center, Spread, and Shape — 6th Math: Statistics & Probability practice questions

Which is a STATISTICAL question?

1. What is my height?
2. What are the heights of students in my class?
3. Is the sky blue?
4. Is 5 even?

Answer: B — What are the heights of students in my class?

Statistical questions anticipate variability — multiple data points. (A), (C), (D) have one answer.

A uniform distribution shows:

1. One peak
2. Bimodal pattern
3. A skewed pattern
4. All values occurring approximately equally — a flat shape

Answer: D — All values occurring approximately equally — a flat shape

Uniform = roughly equal frequencies. (A), (C), (B) describe other shapes.

The MODE of 2, 3, 3, 4, 5, 5, 5 is:

1. 3
2. 5
3. 4
4. 27

Answer: B — 5

Most frequent value. (A), (C), (D) misread.

The mean of 4 numbers is 12. If three of the numbers are 9, 14, and 11, what is the fourth?

1. 14
2. 12
3. 15
4. 13

Answer: A — 14

A) Sum needed = 4 × 12 = 48; known sum = 34; missing = 14. B) The mean. C) Off by one. D) Off by one.

Find the mean of 6, 9, 12, 15, 18.

1. 10
2. 14
3. 12
4. 60

Answer: C — 12

C) Sum = 60, count = 5, 60/5 = 12. A) Off by two. B) Median of similar-looking data. D) That is the sum, not the mean.

A 6th grader claims, 'Median is always the middle value of the list.' This is:

1. Only true if the list is already sorted
2. Always true
3. True only for even-length lists
4. True only when there is a mode

Answer: A — Only true if the list is already sorted

A) You must sort first; 'middle of the list' as written can mislead. B) False without sorting. C) Even-length needs the average of the two middle values after sorting. D) Mode does not control median.

A distribution with most values bunched on the left and a long right tail is:

1. Symmetric
2. Skewed left
3. Skewed right
4. Bimodal

Answer: C — Skewed right

Skew direction = direction of the tail. (A), (B), (D) misread.

The 'spread' (or variability) of a distribution describes:

1. The center
2. How much the data values differ from each other or from the center — measured by range, IQR, or standard deviation
3. The shape only
4. The graph color

Answer: B — How much the data values differ from each other or from the center — measured by range, IQR, or standard deviation

Spread = variation. (A), (C), (D) misread.

Which data set has a mean that EQUALS the median?

1. {1, 1, 1, 1, 10}
2. {1, 2, 3, 9, 10}
3. {2, 4, 6, 8, 10}
4. {0, 0, 0, 0, 25}

Answer: C — {2, 4, 6, 8, 10}

C) Mean and median both equal 6 — symmetric. A) Mean 2.8, median 1. B) Mean 5, median 3. D) Mean 5, median 0.

In a frequency table for {1,1,2,2,2,3,3,4}, the mode is:

1. 1
2. 3
3. 4
4. 2

Answer: D — 2

D) 2 appears three times. A) Twice. B) Twice. C) Once.

Worked example — Find the median of 7, 12, 5, 8, 14.

Step 1: Sort the values. 5, 7, 8, 12, 14. Step 2: Count. n = 5 (odd). Step 3: Pick the middle value (3rd of 5). Answer: median = 8.

What is Q3?

Median of upper half — 75th percentile.

Worked example — Compute the mean from a frequency table: value 2 appears 3 times, 4 appears 5 times, 6 appears 2 times.

Step 1: Multiply each value by its frequency. 2 × 3 = 6. 4 × 5 = 20. 6 × 2 = 12. Step 2: Add the products. 6 + 20 + 12 = 38. Step 3: Add the frequencies. 3 + 5 + 2 = 10. Step 4: Divide. Mean = 38 / 10 = 3.8. Answer: mean = 3.8.

Test scores: 72, 85, 90, 88, 95 — find the mean.

86.