Give Measures of Center and Variability - 6th Grade Math

1. What is the interquartile range (IQR) in terms of $Q_1$ and $Q_3$?

   $\text{IQR}=Q_3-Q_1$. Measures spread of the middle 50% of data.

2. What does a larger MAD tell you about a data set compared to a smaller MAD?

   Values are farther from the mean on average (more variability). Larger MAD indicates data points deviate more from center.

3. Find the mean of the data set $2, 4, 6, 8$.

   $5$. $\frac{2+4+6+8}{4} = \frac{20}{4} = 5$

4. Identify the striking deviation (outlier) in the data set $4,5,5,6,6,7,30$.

   $30$. Much larger than other values ($4-7$).

5. What is the median of a data set after the values are put in order?

   The middle value, or the mean of the two middle values. For odd count, take middle; for even, average the two middle values.

6. What is the median of the ordered data set $2,4,7,9,10$?

   $7$. Middle value of 5 ordered numbers is the 3rd one.

7. What is the median of an ordered data set with an even number of values?

   Average of the two middle values. With even count, median is between the two center values.

8. What does $Q_1$ represent in an ordered data set?

   Median of the lower half of the data. First quartile: 25% of data falls below this value.

9. What is the mean absolute deviation (MAD) from the mean $\bar{x}$ for $n$ values?

   $\text{MAD} = \frac{\sum |x-\bar{x}|}{n}$. Average distance of data points from the mean.

10. A data set has $Q_1=12$ and $Q_3=20$. What is the IQR?

    $8$. $\text{IQR} = Q_3 - Q_1 = 20 - 12 = 8$.

11. What does $Q_3$ represent in an ordered data set?

    Median of the upper half of the data. Third quartile: 75% of data falls below this value.

12. What does a larger IQR indicate about a data set compared to a smaller IQR?

    Greater variability in the middle $50\%$. Larger IQR means more spread in the middle half.

13. What does a larger IQR tell you about a data set compared to a smaller IQR?

    The middle $50\%$ of the data are more spread out. Larger IQR means greater spread in the middle half.

14. Identify $Q_1$ for the ordered data set $2, 3, 5, 8, 9, 10, 12, 15$.

    $4$. Median of lower half: $\{2,3,5,8\}$ is $\frac{3+5}{2}=4$.

15. Find the MAD for the data set $1, 2, 3$ (use mean absolute deviation from the mean).

    $\frac{2}{3}$. Mean is $2$; deviations: $1,0,1$; MAD = $\frac{2}{3}$.

16. Which measure of center is more resistant to outliers: mean or median?

    Median. Outliers affect mean but not median position.

17. Find $Q_3$ for the ordered data $1,2,3,4,5,6,7,8$ using the median-of-halves method.

    $Q_3=\frac{6+7}{2}=6.5$. Median of upper half: $5,6,7,8$.

18. Find the median of the ordered data set $1, 4, 6, 10$.

    $5$. $\frac{4+6}{2} = \frac{10}{2} = 5$ (average of middle two).

19. Which measure of center is usually pulled toward extreme values: mean or median?

    Mean. Extreme values affect the sum, changing the average.

20. What does a larger MAD indicate about a data set compared to a smaller MAD?

    Values are farther from the mean on average. Larger MAD means data is more spread out.

21. What is the interquartile range (IQR) in terms of quartiles $Q_1$ and $Q_3$?

    $\text{IQR} = Q_3 - Q_1$. Measures spread of the middle 50% of data.

22. Which description best indicates a striking deviation: a value far from the rest or a value near the center?

    A value far from the rest (an outlier). Outliers are striking deviations from the pattern.

23. What is the MAD for the data $2,4,6$ (mean $\bar{x}=4$)?

    $\text{MAD}=\frac{|2-4|+|4-4|+|6-4|}{3}=\frac{4}{3}$. Deviations are $2,0,2$; average is $\frac{4}{3}$.

24. Identify the first step to find the median of $6$ numbers.

    Order the data from least to greatest. Must arrange values to find the middle position.

25. Find $Q_1$ for the ordered data $1,2,3,4,5,6,7,8$ using the median-of-halves method.

    $Q_1=\frac{2+3}{2}=2.5$. Median of lower half: $1,2,3,4$.

26. Which measure of center is usually more resistant to outliers: mean or median?

    Median. Outliers don't affect the middle position value.

27. What is the IQR for the ordered data $1,2,3,4,5,6,7,8$ using $Q_1=2.5$ and $Q_3=6.5$?

    $\text{IQR}=6.5-2.5=4$. Subtract first quartile from third quartile.

28. What is the MAD for the data $1,1,1,5$ (mean $\bar{x}=2$)?

    $\text{MAD}=\frac{1+1+1+3}{4}=1.5$. Deviations from $2$ are $1,1,1,3$.

29. Which data set has greater variability by IQR: Set A $\text{IQR}=6$ or Set B $\text{IQR}=2$?

    Set A. Higher IQR indicates more variability.

30. A data set has mean $10$. What is $|x-\bar{x}|$ when a value is $x=6$?

    $4$. Absolute deviation = $|6-10| = 4$.

31. Find the IQR for the ordered data set $2, 3, 5, 8, 9, 10, 12, 15$.

    $7$. $Q_3 - Q_1 = 11 - 4 = 7$ (from previous quartiles).

32. Which data set has greater variability by MAD: Set A $\text{MAD}=1.2$ or Set B $\text{MAD}=2.0$?

    Set B. Higher MAD indicates more variability.

33. Find the median of the ordered data set $3, 7, 9, 12, 20$.

    $9$. Middle value of 5 ordered values is the 3rd one.

34. Identify $Q_3$ for the ordered data set $2, 3, 5, 8, 9, 10, 12, 15$.

    $11$. Median of upper half: $\{9,10,12,15\}$ is $\frac{10+12}{2}=11$.

35. What is the mean of a data set in terms of the total sum $S$ and number of values $n$?

    Mean $= \frac{S}{n}$. Divide the sum of all values by the count of values.

36. What is the median of the ordered data set $3,5,8,12$?

    $\frac{5+8}{2}=6.5$. Even count: average the 2nd and 3rd values.

37. What is the mean absolute deviation (MAD) from the mean for values $x_1,\dots,x_n$?

    $\text{MAD}=\frac{|x_1-\bar{x}|+\cdots+|x_n-\bar{x}|}{n}$. Average distance of all values from the mean.

38. What is the mean of the data set $4,6,8,12$?

    $\frac{4+6+8+12}{4}=7.5$. Sum is $30$, divided by $4$ values.

39. What is the mean of a data set in terms of $n$ values $x_1, x_2, \dots, x_n$?

    $\text{mean}=\frac{x_1+x_2+\cdots+x_n}{n}$. Sum all values and divide by the count.

40. What is the median of an ordered data set with an odd number of values?

    The middle value. With odd count, the median is the single center value.

Study these flashcards