Summary Statistics for a Quantitative Variable - AP Statistics

1. Calculate the weighted mean for $x: 3, 5, 7$ with weights $2, 3, 5$.

   $\bar{x}_w = 5.8$. $\frac{2(3) + 3(5) + 5(7)}{2+3+5} = \frac{58}{10} = 5.8$.

2. Find the z-score for $x = 10$ given $\bar{x} = 8$, $s = 2$.

   $z = 1$. $z = \frac{10-8}{2} = 1$ standard deviation above mean.

3. What does a positive kurtosis indicate?

   Heavier tails than a normal distribution. More extreme values in tails than normal distribution.

4. Determine the median of the dataset: $3, 1, 4, 1, 5$.

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

5. What is the definition of the median?

   The median is the middle value of a dataset when ordered. For odd $n$, it's position $\frac{n+1}{2}$; for even $n$, average of middle two.

6. What is the formula for the weighted mean?

   $\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$. Each value is multiplied by its weight before averaging.

7. Determine the mode of the dataset: $2, 2, 3, 4, 4, 4, 5$.

   Mode = 4. Value $4$ appears three times, more than any other.

8. Define kurtosis in statistics.

   Kurtosis measures the 'tailedness' of the distribution. High kurtosis means heavy tails; low means light tails.

9. What is the formula for population variance?

   $\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2$. Average squared deviation from population mean using $N$.

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

    $z = \frac{x - \bar{x}}{s}$. Number of standard deviations a value is from the mean.

11. Find the variance for dataset: $1, 2, 3, 4, 5$.

    $s^2 = 2.5$. Mean is $3$; variance is $\frac{10}{4} = 2.5$.

12. Identify the formula for coefficient of variation.

    $CV = \frac{s}{\bar{x}} \times 100\%$. Relative variability as a percentage of the mean.

13. Define the term 'outlier' in statistics.

    An outlier is a data point significantly different from others. Falls unusually far from other observations in the dataset.

14. What is the formula for the sample mean of a dataset?

    $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$. Sum all values and divide by sample size $n$.

15. What is skewness in a dataset?

    Skewness measures asymmetry of the probability distribution. Positive skew has right tail; negative skew has left tail.

16. Identify the formula for variance of a sample.

    $s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2$. Average squared deviation from sample mean using $n-1$.

17. Define the interquartile range (IQR).

    IQR = Q3 - Q1. Distance between third quartile and first quartile.

18. Identify the formula for the sample standard deviation.

    $s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}$. Square root of sample variance using $n-1$ in denominator.

19. Identify the formula for the sample covariance.

    $s_{xy} = \frac{1}{n-1} \sum (x_i - \bar{x})(y_i - \bar{y})$. Measures how two variables change together linearly.

20. Find the percentile rank of $4$ in $1, 2, 4, 5, 6$.

    60th percentile. Three out of five values are below $4$.

21. What does a boxplot visually represent?

    A boxplot displays the distribution through five-number summary. Shows quartiles, median, and potential outliers graphically.

22. Find the range of the dataset: $9, 4, 7, 2, 5$.

    Range = 7. Maximum $9$ minus minimum $2$ equals $7$.

23. What is the definition of a percentile?

    A percentile indicates the relative standing of a value in a dataset. Shows what percentage of data falls below a given value.

24. What is the median of $8, 3, 6, 7, 5$?

    Median = 6. Ordered: $3, 5, 6, 7, 8$; middle value is $6$.

25. What is a histogram used for?

    A histogram displays the frequency distribution of a dataset. Shows data distribution using bars for frequency counts.

26. Calculate the range for $10, 15, 20, 25, 30$.

    Range = 20. Maximum $30$ minus minimum $10$ equals $20$.

27. Calculate the mean of the dataset: $2, 4, 6, 8, 10$.

    $\bar{x} = 6$. Sum is $30$, divided by $5$ observations gives $6$.

28. Identify the IQR of the dataset: $1, 3, 3, 6, 7, 8, 9$.

    IQR = 5. $Q^3 = 8$, $Q^1 = 3$, so $IQR = 8 - 3 = 5$.

29. State the formula for the population mean.

    $\mu = \frac{1}{N} \sum_{i=1}^{N} x_i$. Sum all values and divide by population size $N$.

30. What is the skewness of a symmetric distribution?

    Skewness = 0. Symmetric distributions have equal left and right tails.

31. Calculate the standard deviation of $4, 4, 4, 4, 4$.

    $s = 0$. All values are identical, so no variability exists.

32. How is the range of a dataset calculated?

    Range = Maximum value - Minimum value. Difference between highest and lowest values in the dataset.

33. What is the five-number summary in statistics?

    Minimum, Q1, Median, Q3, Maximum. Five key values that describe a dataset's distribution.

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