Center, Shape, & Spread of Data - SAT Math

1. Calculate the variance: data set 2, 2, 3, 3.

   Variance = 0.25. Mean is 2.5, deviations squared and averaged.

2. What is the median of the data set: 3, 7, 9, 2, 4?

   The median is 4. Middle value when ordered: 2, 3, 4, 7, 9.

3. What does the skewness of a distribution indicate?

   Skewness indicates the asymmetry of a distribution. Measures if distribution leans left or right.

4. Find the mean: 4, 8, 6, 5.

   Mean = 5.75. $(4+8+6+5) ÷ 4 = 5.75$

5. What is an outlier in a data set?

   An outlier is a data point significantly different from others. Falls far from typical data pattern.

6. What is the purpose of a stem-and-leaf plot?

   It displays quantitative data to show distribution details. Separates tens and ones digits clearly.

7. Find the mode: 3, 3, 6, 6, 6, 2, 1.

   Mode = 6. 6 appears most frequently (3 times).

8. Identify the best measure of center for skewed data.

   The median is the best measure for skewed data. Outliers don't affect median much.

9. Find the interquartile range: Q1 = 5, Q3 = 15.

   IQR = 10. $Q^3 - Q^1 = 15 - 5 = 10$

10. Find the range: data set 8, 8, 9, 10, 10.

    Range = 2. $10 - 8 = 2$

11. What is a cumulative frequency graph?

    It shows cumulative totals of data points up to each class. Running total increases at each point.

12. State the formula for variance.

    $Variance = \frac{\sum (x_i - \text{mean})^2}{n}$. Square deviations from mean, then average.

13. What is the range of the data set: 8, 14, 3, 20, 7?

    Range = 17. Difference between maximum and minimum values: $20 - 3 = 17$.

14. What does a box plot display?

    A box plot displays the distribution, median, and quartiles of a data set. Five-number summary in visual format.

15. Identify the shape of a symmetric distribution.

    A symmetric distribution is mirrored around its center. Both sides look identical when folded.

16. State the formula for the z-score of a data point.

    $z = \frac{x - \text{mean}}{\text{standard deviation}}$. Standardizes distance from mean in units.

17. Identify the median in the data set: 3, 5, 7, 9, 11.

    Median = 7. Middle value when data is ordered from least to greatest.

18. What is the mean of the data set: 4, 8, 6, 10, 2?

    The mean is 6. Sum all values (30) and divide by count (5).

19. How does a dot plot display data?

    A dot plot uses dots to represent frequency of data values. Each dot equals one data point.

20. Which shape of distribution is symmetric with one peak?

    Normal distribution. Bell-shaped curve with mean, median, and mode all equal.

21. Find the interquartile range (IQR) for Q1 = 25 and Q3 = 75.

    IQR = 50. Difference between third and first quartiles: $75 - 25 = 50$.

22. Find the z-score: x = 10, mean = 8, std dev = 1.

    z = 2. $(10-8) ÷ 1 = 2$ standard deviations.

23. What are quartiles?

    Quartiles divide data into four equal parts. 25%, 50%, 75% split points.

24. What is the interquartile range (IQR)?

    IQR = Q3 - Q1. Third quartile minus first quartile.

25. State the formula for the variance of a data set.

    Variance = $\frac{\text{Sum of squared deviations from mean}}{\text{Number of data points}}$. Measures spread by averaging squared distances from mean.

26. Identify the effect of skewness on mean and median.

    In skewed data, mean is pulled toward the tail. Median stays centered, mean shifts.

27. Calculate the range: 12, 7, 19, 5, 10.

    Range = 14. $19 - 5 = 14$

28. Identify the shape of a distribution with a long right tail.

    The shape is right-skewed. Long tail extends toward higher values on the right.

29. How is the geometric mean calculated?

    $Geometric mean = \sqrt[n]{x_1 \cdot x_2 \cdot \dots \cdot x_n}$. Multiply all values, take nth root.

30. What is a bimodal distribution?

    A distribution with two distinct peaks or modes. Two separate frequency peaks exist.

31. Describe a negatively skewed distribution.

    Negatively skewed: tail extends to the left. Most data on right, few low values.

32. Calculate the mean absolute deviation: 2, 4, 4, 6.

    Mean absolute deviation = 1. Average distance from mean is 1.

33. What is the mode of a data set?

    The mode is the most frequently occurring value. Count frequency of each value.

34. Describe a positively skewed distribution.

    Positively skewed: tail extends to the right. Most data on left, few high values.

35. How is the median of a data set defined?

    The median is the middle value when data is ordered. Sort values first, then find center.

36. What does a scatter plot show?

    A scatter plot shows the relationship between two variables. Points reveal correlation patterns.

37. How is range calculated in a data set?

    Range = Maximum value - Minimum value. Subtract smallest from largest value.

38. What does the term 'spread of data' refer to?

    It refers to the variability or dispersion of data points. How scattered data points are.

39. Find the standard deviation: data set 4, 4, 5, 5.

    Standard deviation = 0.5. Mean is 4.5, deviations squared and averaged.

40. What is a histogram used for?

    A histogram displays the frequency distribution of data. Bars show count in each interval.

41. What does a frequency polygon represent?

    It represents the frequencies of data points in a line graph. Connected dots show frequency distribution.

42. Which measure of center is most affected by outliers?

    The mean is most affected. Outliers pull the mean toward extreme values.

43. What is the formula for the z-score of a data point?

    z = $\frac{x - \text{mean}}{\text{standard deviation}}$. Standardizes values by measuring distance from mean in standard deviations.

44. What is the standard deviation if variance is 16?

    Standard Deviation = 4. Standard deviation is the square root of variance: $\sqrt{16} = 4$.

45. What is the shape of a uniform distribution?

    A uniform distribution has equal frequency across values. All intervals have same probability.

46. What does the standard deviation measure?

    The standard deviation measures data spread around the mean. Shows how far values deviate from center.

47. What is the mode of the data set: 4, 4, 5, 6, 6, 6, 7?

    Mode = 6. Value that appears most frequently in the data set.

48. Find the median: 3, 1, 4, 2, 5.

    Median = 3. Ordered: 1,2,3,4,5; middle is 3.

49. What is the range of a data set?

    The range is the difference between the highest and lowest values. Span from minimum to maximum value.

50. Find the median: data set 5, 9, 3.

    Median = 5. Ordered: 3,5,9; middle value is 5.

51. What is the formula for the mean of a data set?

    $Mean = \frac{\sum x_i}{n}$. Sum all values, divide by count.

52. Identify the measure of central tendency least affected by outliers.

    The median is least affected by outliers. Extreme values don't shift median much.

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