Descriptive Statistics (Mean, Median, SD) - Quantitative Reasoning

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

   $Q_3-Q_1$. The IQR captures the middle 50% spread by differencing the third and first quartiles, excluding extremes.

2. What is the mean after adding a constant $c$ to every data value?

   New mean $=\bar{x}+c$. Adding a constant shifts the entire data set equally, increasing the mean by that constant.

3. What is the outlier rule using $Q_1$, $Q_3$, and $\text{IQR}$ (Tukey fences)?

   Outliers $<Q_1-1.5\,\text{IQR}$ or $>Q_3+1.5\,\text{IQR}$. Tukey's fences identify outliers as values beyond 1.5 IQR from the quartiles, flagging extreme deviations.

4. What happens to the standard deviation when every data value is multiplied by $k$?

   New SD $=|k|\sigma$. Multiplying by k scales the deviations, so the standard deviation adjusts by the absolute value of k.

5. What is the formula for the arithmetic mean of $n$ numbers $x_1,\dots,x_n$?

   $\bar{x}=\frac{x_1+\cdots+x_n}{n}$. The arithmetic mean represents the average by summing all observations and dividing by their total number.

6. Identify the median of the ordered data set $\{1,3,7,9\}$.

   $5$. For even n=4, averaging the second and third values (3 and 7) gives the median.

7. Which measure of spread is most resistant to extreme outliers: range, IQR, or SD?

   IQR. IQR focuses on the central 50% of data, making it robust against outliers that affect range and SD.

8. What is the median of an ordered data set with an even number $n$ of values?

   Average of positions $\frac{n}{2}$ and $\frac{n}{2}+1$. For even n, the median averages the two central values to represent the middle of the ordered data set.

9. Identify the new mean if the mean is $12$ and each value increases by $5$.

   $17$. Adding 5 to each value shifts the original mean of 12 by the same amount.

10. What is the weighted mean of values $x_i$ with weights $w_i$?

    $\frac{\sum w_i x_i}{\sum w_i}$. The weighted mean accounts for varying importance by multiplying each value by its weight and normalizing by total weight.

11. What is the standard deviation after adding a constant $c$ to every data value?

    Unchanged. Adding a constant preserves the spread, as deviations from the new mean remain identical.

12. What does a z-score of $z=0$ indicate about the value $x$?

    $x$ equals the mean. A z-score of zero signifies that the value is exactly at the distribution's mean, with no deviation.

13. What is the range of a data set in terms of its maximum and minimum?

    $\max-\min$. The range measures the total spread by subtracting the smallest value from the largest in the data set.

14. What is the mode of a data set?

    The most frequently occurring value. The mode identifies the value that appears most often, indicating the peak frequency in the data distribution.

15. State the population variance formula for data $x_1,\dots,x_n$ with mean $\bar{x}$.

    $\sigma^2=\frac{1}{n}\sum_{i=1}^n(x_i-\bar{x})^2$. Population variance averages the squared deviations from the mean over all n observations.

16. What is the z-score of a value $x$ given mean $\mu$ and standard deviation $\sigma$?

    $z=\frac{x-\mu}{\sigma}$. The z-score standardizes a value by measuring its distance from the mean in standard deviation units.

17. Find the z-score of $x=86$ for a distribution with $\mu=80$ and $\sigma=3$.

    $2$. Subtract mean from x and divide by SD: (86-80)/3 standardizes the value.

18. What is the median of an ordered data set with an odd number $n$ of values?

    The middle value, at position $\frac{n+1}{2}$. In an ordered list with odd n, the median is the central value that divides the data into two equal parts.

19. What is the definition of the $p$th percentile of a data set?

    A value with about $p\%$ of data at or below it. The pth percentile divides the ordered data such that p% falls at or below it, marking relative position.

20. Identify the mean of the data set $\{2,4,6,8\}$.

    $5$. Summing the values 2+4+6+8=20 and dividing by 4 yields the arithmetic mean of the set.

21. What is the standard deviation in terms of the variance $\sigma^2$?

    $\sigma=\sqrt{\sigma^2}$. Standard deviation is the square root of variance, quantifying the average deviation from the mean.

22. Which measure is most resistant to extreme outliers: mean, median, or mode?

    Median. The median, as the middle value, remains unaffected by extreme outliers unlike the mean or mode.

23. What is the mean after multiplying every data value by a constant $k$?

    New mean $=k\bar{x}$. Multiplying by k scales the data linearly, thus scaling the mean by the same factor.

24. Identify the IQR for the ordered data set $\{1,2,3,4,5,6,7,8\}$ using medians of halves.

    $4$. Q1 is the median of the lower half (2.5), Q3 of the upper (6.5), so IQR=6.5-2.5.

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