Assess Visual Overlap of Distributions - 7th Grade Math

1. Identify the correct conclusion if $|\Delta|=2$ and $\text{MAD}=8$ for two groups.

   Strong overlap because $k=\frac{2}{8}=0.25$ is small. $k=\frac{2}{8}=0.25$ is close to 0, indicating minimal separation.

2. What does it mean to compare two distributions by expressing center difference as a multiple of variability?

   Compute $\frac{|\text{center}_1-\text{center}_2|}{\text{variability}}$ and interpret the size. Divide center difference by variability to quantify separation.

3. What is the formula for mean absolute deviation (MAD) from the mean?

   $\text{MAD}=\frac{\sum |x-\bar{x}|}{n}$. Average the absolute deviations from the mean.

4. Find the separation multiple if $\bar{x}_A=18$, $\bar{x}_B=12$, and $\text{MAD}=2$.

   $3$. $\frac{|18-12|}{2}=\frac{6}{2}=3$

5. What does $k=\frac{|\Delta|}{\text{MAD}}\approx 2$ suggest about overlap?

   Noticeable separation; overlap is smaller. When $k=2$, centers differ by twice the typical spread.

6. If two groups differ by $12$ and the common MAD is $3$, what is the separation in MADs?

   $4$. Divide difference by MAD: $\frac{12}{3}=4$.

7. Find the separation multiple if $\bar{x}_A=70$, $\bar{x}_B=64$, and $\text{MAD}=3$.

   $2$. $\frac{|70-64|}{3}=\frac{6}{3}=2$

8. Which statement best matches "similar variabilities" for two groups?

   Their MAD values are about the same size. Similar variabilities means the spreads are approximately equal.

9. What does a separation multiple of at least $3$ usually suggest about overlap?

   Little overlap; centers are far apart compared with variability. Large multiples indicate minimal distribution overlap.

10. What measure of variability is named in CCSS.7.SP.3 as an example?

    Mean absolute deviation (MAD). MAD measures average distance from the mean, showing spread.

11. If $\Delta=12$ and $k=3$, what is $\text{MAD}$ from $k=\frac{|\Delta|}{\text{MAD}}$?

    $\text{MAD}=4$. Rearrange: $\text{MAD}=\frac{|\Delta|}{k}=\frac{12}{3}=4$

12. Which option indicates the least overlap: multiple $0.5$, $1$, $2$, or $3$?

    $3$. Largest multiple means greatest separation.

13. Identify the best interpretation of $\frac{|\bar{x}_A-\bar{x}_B|}{\text{MAD}}=2$.

    The mean difference is about $2$ times the typical distance from the mean. Centers differ by twice the typical deviation.

14. What does a separation multiple near $0$ suggest about visual overlap on a dot plot?

    Heavy overlap; centers are very close compared with variability. Small multiples mean centers are close relative to spread.

15. If $|bar{x}_1-bar{x}_2|=8$ and $MAD=4$, how many MADs apart are the means?

    $2$. Divide difference by MAD: $\frac{8}{4}=2$.

16. What is the "center" of a numerical distribution in this standard?

    A typical value such as the mean or the median. The center represents where most data values cluster in a distribution.

17. Which option indicates the most overlap: multiple $0.2$, $1.4$, $2.1$, or $3.0$?

    $0.2$. Smallest multiple means greatest overlap.

18. Identify the correct unit for $k=\frac{|\Delta|}{\text{MAD}}$ (units or unitless).

    Unitless (the units cancel). Both numerator and denominator have same units, which cancel out.

19. Choose the correct conclusion if $\frac{|c_1-c_2|}{v}=2$ and spreads are similar.

    Separation is noticeable; overlap is reduced. Ratio of 2 shows clear separation with limited overlap.

20. What is the "difference in centers" if the means are $bar{x}_1=72$ and $bar{x}_2=68$?

    $4$. Subtract means: $|72-68|=4$.

21. Choose the correct conclusion if $rac{|c_1-c_2|}{v}=0.5$ and spreads are similar.

    Expect substantial overlap between the two distributions. Ratio < 1 indicates distributions mostly overlap.

22. Find $\text{MAD}$ if $|\bar{x}_A-\bar{x}_B|=8$ and the multiple is $2$.

    $4$. $\text{MAD}=\frac{8}{2}=4$ since $8=2\times\text{MAD}$

23. Find the multiple of variability: $|c_1-c_2|=10$ and $v=5$. What is $rac{|c_1-c_2|}{v}$?

    $2$. Divide difference by variability: $\frac{10}{5}=2$.

24. If $\text{MAD}=5$ and $k=1.2$, what is $|\Delta|$ from $k=\frac{|\Delta|}{\text{MAD}}$?

    $|\Delta|=6$. Rearrange: $|\Delta|=k \times \text{MAD}=1.2 \times 5=6$

25. What does a separation multiple around $2$ suggest about visual overlap on a dot plot?

    Noticeable separation; overlap is reduced compared with variability. Centers separated by 2 MADs show clear distinction.

26. Find $k$ if $\Delta=6$ and $\text{MAD}=3$ using $k=\frac{|\Delta|}{\text{MAD}}$.

    $k=2$. $k=\frac{|6|}{3}=\frac{6}{3}=2$

27. What measure of center is most commonly used in CCSS.7.SP.3 comparisons?

    The mean (or sometimes the median) of each distribution. Mean is the primary center measure for this standard.

28. What is the formula for the mean difference in centers between two groups $A$ and $B$?

    $\Delta=|\bar{x}_A-\bar{x}_B|$. Absolute value ensures positive distance between means.

29. Find $|\bar{x}_A-\bar{x}_B|$ if $\bar{x}_A=25$ and $\bar{x}_B=31$.

    $6$. $|25-31|=|-6|=6$

30. Find the multiple of variability: $|c_1-c_2|=6$ and $v=3$. What is $rac{|c_1-c_2|}{v}$?

    $2$. Divide difference by variability: $\frac{6}{3}=2$.

31. Find $\Delta$ if $\bar{x}_A=72$ and $\bar{x}_B=68$ using $\Delta=\bar{x}_A-\bar{x}_B$.

    $\Delta=4$. $\Delta=72-68=4$

32. What measure of variability is named in CCSS.7.SP.3 as a typical choice?

    Mean absolute deviation (MAD). MAD measures average distance from mean, good for spread.

33. What does $k=\frac{|\Delta|}{\text{MAD}}\approx 0$ suggest about overlap?

    Heavy overlap; centers are nearly the same. When $k$ is near 0, the difference is tiny relative to spread.

34. Which condition justifies using one MAD value for both groups in a comparison?

    The two distributions have similar variability (similar MAD or spread). Similar MADs allow using one value for both groups.

35. Choose the pair with more overlap: Pair A $k=\frac{3}{6}$ or Pair B $k=\frac{6}{3}$.

    Pair A, because $\frac{3}{6}=0.5$ is smaller than $\frac{6}{3}=2$. Smaller $k$ means more overlap: $0.5 < 2$.

36. Find $k$ if $\Delta=4$ and $\text{MAD}=8$ using $k=\frac{|\Delta|}{\text{MAD}}$.

    $k=0.5$. $k=\frac{|4|}{8}=\frac{4}{8}=0.5$

37. What does a larger value of $k=\frac{|\Delta|}{\text{MAD}}$ indicate?

    Less overlap and a stronger difference between groups. Higher $k$ means centers are farther apart relative to variability.

38. What ratio is used to express separation when two groups have similar variability?

    $\text{separation multiple}=\frac{|\bar{x}_A-\bar{x}_B|}{\text{MAD}}$. Ratio shows how many MADs separate the centers.

39. Find the multiple of variability: $|c_1-c_2|=3$ and $v=6$. What is $\frac{|c_1-c_2|}{v}$?

    $0.5$. Divide difference by variability: $\frac{3}{6}=0.5$.

40. What does "visual overlap" of two distributions describe on a dot plot?

    How much the dot patterns cover the same range of values. Visual overlap shows shared value ranges between distributions.

41. Compute $\text{MAD}$ for data $\{2,4,6\}$ using $\text{MAD}=\frac{\sum |x-\bar{x}|}{n}$.

    $\frac{4}{3}$. $\bar{x}=4$; deviations: $2,0,2$; MAD=\frac{4}{3}$

42. What measure of center is most commonly compared in CCSS.7.SP.3 examples?

    The mean. Mean is the most common center measure for comparing distributions.

43. Find the multiple if $|\bar{x}_A-\bar{x}_B|=10$ and $\text{MAD}=5$.

    $2$. $\frac{10}{5}=2$

44. What is the main reason to compare $|c_1-c_2|$ to $v$ instead of using $|c_1-c_2|$ alone?

    It scales the difference by typical spread, making the comparison meaningful. Standardizing allows comparison across different data scales.

45. Choose the larger separation: Pair 1 $k=\frac{8}{4}$ or Pair 2 $k=\frac{9}{6}$.

    Pair 1, because $\frac{8}{4}=2$ is larger than $\frac{9}{6}=1.5$. Compare: $\frac{8}{4}=2$ vs $\frac{9}{6}=1.5$; 2 > 1.5.

46. Compute $\text{MAD}$ for data $\{5,5,9\}$ using $\text{MAD}=\frac{\sum |x-\bar{x}|}{n}$.

    $\frac{16}{9}$. $\bar{x}=\frac{19}{3}$; sum of absolute deviations is $\frac{16}{3}$

47. If $\bar{x}_A=52$, $\bar{x}_B=47$, and $MAD=2.5$, what is $\frac{|\bar{x}_A - \bar{x}_B|}{MAD}$?

    $2$. Calculate $\frac{|52-47|}{2.5}=\frac{5}{2.5}=2$.

48. Which condition is assumed when comparing overlap using multiples of variability in CCSS.7.SP.3?

    The two distributions have similar variability (similar spread). Similar spread allows meaningful comparison of separation.

49. Identify the correct comparison: difference in means $=9$, MAD $=3$. Which statement matches?

    The means are about $3$ MADs apart. Calculate $\frac{9}{3}=3$ MADs separation.

50. Find $k$ if $\Delta=10$ and $\text{MAD}=5$ using $k=\frac{|\Delta|}{\text{MAD}}$.

    $k=2$. $k=\frac{|10|}{5}=\frac{10}{5}=2$

51. What measure of variability is named in CCSS.7.SP.3 for comparing distributions?

    Mean absolute deviation (MAD). MAD measures average distance from the mean.

52. If $\bar{x}_A=15$, $\bar{x}_B=20$, what is $|\Delta|$ for $\Delta=\bar{x}_A-\bar{x}_B$?

    $|\Delta|=5$. $\Delta=15-20=-5$, so $|\Delta|=|-5|=5$

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