Statistics is a measure taken from the sample to help us analyze the data. Meanwhile, a parameter is the measure taken from the population. In inferential statistics, we will use statistics to make inferences about the parameters. For now, we'll focus on summary statistics. Mean, median, standard deviation, IQR, range, all are summary statistics for a quantitative variable.
- The mean, median, quartiles, and percentiles measure the center and position for quantitative data
- The range, IQR, and standard deviation measure the variability for quantitative data.
The summary measures change if we convert them to different units.
Statistics of Center
The Mean
Mean, or average, as you learned before, is easy to calculate, we add up all the values of the variable and divide the sum by number. The formula follows: x̄ = ∑x / n x is read as an x bar; it’s the mean value of the x values of data. By the way, it doesn't need to be x; it can be y as well. Means are the best summary measures for a symmetric distribution because, as mentioned before, they are the balancing point of the distributions. However, the mean has few drawbacks.It does not tell about all individuals (that is why we also need summary measures of spread), and it is not resistant to outliers.
The mean number can easily be affected by one high value in our data set and affect our study results, leading us to make wrong decisions if we wrongly choose to report the mean instead of the median.
The Median
Median is the middle number of data. When data are even we calculate the median by finding the average of the middle two numbers. Medians are good alternatives of summarizing the center of for skewed distributions or distribution with an outlier. The median is resistant to outliers. However, it is not easy to find the median from the histogram, but you don’t need to do it.
We will need only to find its position by dividing the total number of our data by 2. If the total amount is odd, we add one (n/2 for even cases and (n + 1)/2 for odd ones).
In the following section, when we compare two histograms, you will see how to find the median from the histogram.
Mean or Median?
Rule of thumb time!
If the distribution is symmetric and unimodal, the mean is often the best measure of central tendency because it takes into account all of the values in the dataset and reflects the overall trend in the data.
On the other hand, if the distribution is skewed or has outliers, the median is often a better measure of central tendency because it is resistant to the influence of these values. In right-skewed distributions, the mean is generally higher than the median, while in left-skewed distributions, the mean is generally lower than the median.
It's always a good idea to report both the mean and median when describing the statistical properties of a dataset, and to explain why they are different if they are not close to each other. This can help to provide a more complete picture of the distribution of the data and how it is dispersed around the center.
Likewise, remember to always report the units when describing summary measures of the center, as you would in any math class. This helps to provide context and allows others to interpret the results accurately.
Statistics of Spread
Standard Deviation
The standard deviation is like lungs in statistics. You cannot breathe without it. You cannot analyze data without it. It shows how far or close the values are dispersed, deviated, or vary from the mean. The process of calculating standard deviation is lengthy and time-consuming, and definitely, you already know by now. You will mostly rely on your calculator to do it for you, but in case here is the formula:
s = √[∑(x-x̄)^2/n-1]
You may wonder, if not already before, why subtract one from n? When calculating the standard deviation for a sample, it is necessary to subtract 1 from the number of values (n) in the sample to account for the fact that the sample is a subset of the population and therefore has some level of sampling error. This is known as the "degrees of freedom" and it is used to adjust the variance estimate so that it is more accurate and more representative of the population.
As you read more units, you will revisit the concept of standard deviation and will understand it more.
Interquartile Range (IQR)
Recall that the interquartile range (IQR) is based on the difference between the upper and lower quartiles. It is calculated as the difference between the 75th percentile (Q3) and the 25th percentile (Q1) in a data set.
IQR = upper quartile (Q3) - lower quartile (Q1).
The first quartile, Q1, is the median of the half of the ordered data set from the minimum to the position of the median. The third quartile, Q3, is the median of the half of the ordered data set from the position of the median to the maximum. Q1 and Q3 form the boundaries for the middle 50% of values in an ordered data set.
However, the IQR does not capture the entire distribution of values in the data set and therefore may not fully reflect the variability of the data. Other measures such as the range, standard deviation, and variance can provide a more comprehensive view of the dispersion and variability in a data set. These measures are often used in conjunction with the IQR to provide a more complete understanding of the characteristics of a data set.
Standard Deviation or IQR?
It's generally true that the IQR is larger than the standard deviation for symmetric distributions without outliers, although the specific relationship between these measures will depend on the characteristics of the data set.
- For a symmetric, unimodal distribution, the mean and median will be approximately equal, and the standard deviation and IQR will provide complementary information about the dispersion of the data. In this case, it is appropriate to report both the mean and standard deviation to provide a sense of the center and spread of the distribution.
- For skewed distributions, the median is often a better measure of central tendency than the mean, as the mean can be influenced by extreme values or outliers. In this case, it is appropriate to report both the median and IQR to provide a sense of the center and spread of the distribution.
In general, report both measures of center and spread together is a good plan-of-action, as this provides a more complete understanding of the characteristics of a data set. Reporting only one measure, such as the standard deviation or IQR, can be misleading or incomplete, as it does not provide a full picture of the data.
A Note About Outliers
Previously, we've talked about what outliers are, but how do we know a data point is an outlier or not? There are many methods for determining outliers. Two methods frequently used in this course are:
Method I: 1.5 x IQR
We can use the IQR to identify outliers involves calculating the IQR for the data set and then using this value to determine which values are outside the normal range of the data.
Specifically, values that are more than 1.5 × IQR above the third quartile (Q3) or more than 1.5 × IQR below the first quartile (Q1) are considered outliers. This method is based on the assumption that most of the values in the data set should fall within the range defined by the IQR, with only a small number of values falling outside this range.
Example
To determine whether a value is an outlier using the 1.5 × IQR method, you will need to calculate the interquartile range (IQR) for the data set and then compare the value to the upper and lower bounds of the data set. Here is an example of how this might be done:
Suppose you have the following data set: 10, 15, 20, 25, 30, 35, 40, 45, 50
To determine whether any of these values are outliers using the 1.5 × IQR method, you would first need to calculate the IQR. To do this, you would need to find the first quartile (Q1), the median (Q2), and the third quartile (Q3).
For this data set, the first quartile (Q1) is 20, the median (Q2) is 30, and the third quartile (Q3) is 40. The IQR is then calculated as the difference between Q3 and Q1, or 40 - 20 = 20.
Next, you would need to determine the upper and lower bounds of the data set using the IQR. The upper bound is calculated as Q3 + 1.5 × IQR, or 40 + (1.5 × 20) = 70. The lower bound is calculated as Q1 - 1.5 × IQR, or 20 - (1.5 × 20) = -10.
Finally, you would need to compare the value you are interested in to these bounds. If the value is less than the lower bound or greater than the upper bound, it is considered an outlier. For example, if the value you are interested in is 100, it is an outlier because it is greater than the upper bound of 70. If the value you are interested in is 5, it is not an outlier because it is within the bounds of the data set (-10 to 70).
Method II: Standard Deviations
We can also use standard deviations to identify outliers involves calculating the mean and standard deviation for the data set and then using these values to determine which values are outside the normal range of the data. Specifically, values that are more than 2 standard deviations above or below the mean are considered outliers. This method is based on the assumption that most of the values in the data set should fall within two standard deviations of the mean, with only a small number of values falling outside this range.
Both of these methods can be useful for identifying unusual or unexpected values in a data set, but they may not be suitable for all types of data or in all situations. It is important to consider the characteristics of the data set and the goals of the analysis when deciding which method to use to identify outliers.
Resistance and Nonresistant Measures
The mean, standard deviation, and range are considered nonresistant (or non-robust) because they are influenced by outliers. The median and IQR are considered resistant (or robust), because outliers do not greatly (if at all) affect their value.
For these reasons, the median and IQR are often preferred to the mean, standard deviation, and range when working with data sets that may contain outliers. They are more robust and provide a more accurate representation of the center and spread of the data, even in the presence of extreme values.
Key Vocabulary
- Mean
- Median
- Mode
- Range
- IQR
- Standard Deviation
- Outliers
🎥 Watch: AP Stats - Unit 1 Streams
Vocabulary
The following words are mentioned explicitly in the College Board Course and Exam Description for this topic.
| Term | Definition |
|---|---|
| first quartile | The median of the lower half of an ordered data set, denoted as Q1, marking the boundary below which 25% of the data falls. |
| interquartile range | A measure of variability calculated as the difference between the third quartile (Q3) and the first quartile (Q1), representing the spread of the middle 50% of data. |
| mean | The average value of a dataset, represented by μ in the context of a population. |
| measures of center | Numerical summaries that describe the central tendency of a data set, including the mean and median. |
| measures of position | Numerical summaries that describe the location of data values within a distribution, including quartiles and percentiles. |
| measures of variability | Statistical measures that describe how spread out or dispersed data values are in a distribution. |
| median | The middle value when data are ordered; for an even number of data points, typically the average of the two middle values. |
| nonresistant | A characteristic of a statistic that is significantly affected or influenced by outliers; also called non-robust. |
| outlier | Data points that are unusually small or large relative to the rest of the data. |
| percentile | A value such that p% of the data is less than or equal to it, used to describe the position of a data point within a distribution. |
| Q1 | The first quartile; the value below which 25% of the data falls. |
| Q3 | The third quartile; the value below which 75% of the data falls. |
| quartile | A value that divides an ordered data set into four equal parts; Q1 and Q3 form the boundaries for the middle 50% of values. |
| range | A measure of variability calculated as the difference between the maximum and minimum data values in a dataset. |
| resistant | A characteristic of a statistic that is not greatly affected by outliers; also called robust. |
| sample standard deviation | The standard deviation calculated for a sample, denoted by s, using the formula s = √(1/(n-1) ∑(xᵢ-x̄)²). |
| sample variance | The square of the sample standard deviation, denoted by s², representing variability in squared units. |
| standard deviation | A measure of how spread out data values are from the mean, represented by σ in the context of a population. |
| statistic | Numerical summaries or measures calculated from sample data, such as mean, median, or standard deviation. |
| third quartile | The median of the upper half of an ordered data set, denoted as Q3, marking the boundary below which 75% of the data falls. |
Frequently Asked Questions
How do I calculate the mean when I have a bunch of data points?
The mean (sample mean) is the average—add all your data values and divide by how many there are. In AP stats notation: x̄ = (1/n) Σ xi. Steps: 1) Sum every data point. 2) Count n (the number of values). 3) Divide the sum by n. Example: data = {4, 7, 9, 10}. Sum = 30, n = 4, so x̄ = 30/4 = 7.5. A few quick AP notes from Topic 1.7: the mean is nonresistant (outliers can pull it a lot), so use the median/IQR when data are skewed (CED UNC-1.I and UNC-1.K). The formula and related formulas (like sample standard deviation s) are on the AP formula sheet, and you can use a graphing calculator on the exam (see unit resources). For a refresher and practice problems, check the Topic 1.7 study guide (https://library.fiveable.me/ap-statistics/unit-1/summary-statistics-for-quantitative-variable/study-guide/fDwLeu9W74iSnEcnKHOA), the Unit 1 overview (https://library.fiveable.me/ap-statistics/unit-1), and practice sets (https://library.fiveable.me/practice/ap-statistics).
What's the formula for standard deviation and why is there an n-1 in the denominator?
The sample standard deviation formula (from the AP formula sheet) is s = sqrt[ (1/(n − 1)) · Σ(xi − x̄)² ]. We square deviations from the sample mean x̄, sum them, divide by n − 1, then take the square root. The squared quantity s² is the sample variance. Why n − 1? Because x̄ is itself estimated from the same data, it “uses up” one degree of freedom. Dividing by n would systematically give a number that’s too small (it underestimates the true population variance). Using n − 1 corrects that bias so s² is an unbiased estimator of the population variance. In short: one parameter (the mean) is estimated from the data, so only n − 1 independent deviations remain. For AP review, see the Topic 1.7 study guide (https://library.fiveable.me/ap-statistics/unit-1/summary-statistics-for-quantitative-variable/study-guide/fDwLeu9W74iSnEcnKHOA) and try practice problems at (https://library.fiveable.me/practice/ap-statistics).
When do I use median vs mean to describe the center of my data?
Use the mean when the distribution is roughly symmetric and has no outliers—it summarizes every value (x̄ = Σxi/n) and pairs naturally with the standard deviation. Use the median when the distribution is skewed or has outliers because the median (middle ordered value) is resistant and won’t shift a lot if a few values are extreme. Quick rules you’ll use on the AP: if the histogram/boxplot is symmetric, report mean ± standard deviation; if it’s skewed or has outliers (check 1.5×IQR rule or values ±2 standard deviations), report median and IQR. Be ready to justify your choice on the exam by describing shape, outliers, and resistance (CED UNC-1.K). For a short review, see the Topic 1.7 study guide (https://library.fiveable.me/ap-statistics/unit-1/summary-statistics-for-quantitative-variable/study-guide/fDwLeu9W74iSnEcnKHOA). For extra practice, try problems at (https://library.fiveable.me/practice/ap-statistics).
What's the difference between Q1 and Q3 and how do I find them?
Q1 and Q3 are the first and third quartiles—they mark the lower and upper boundaries of the middle 50% of your data (UNC-1.I.4). Q1 is the median of the lower half of the ordered data (from the minimum up to the overall median); Q3 is the median of the upper half (from the overall median up to the maximum). To find them: - Order the data. - Find the median. If n is odd, don’t include the median in the halves; if n is even, split evenly. - Q1 = median of the lower half; Q3 = median of the upper half. Example: for 9 ordered values, median is 5th; Q1 is median of values 1–4 (average of 2nd & 3rd if needed), Q3 is median of values 6–9. Use Q3−Q1 to get IQR (UNC-1.J.2), and flag outliers beyond Q3 + 1.5·IQR or below Q1 − 1.5·IQR (UNC-1.K.1). For more practice and AP-aligned review, see the Topic 1.7 study guide (https://library.fiveable.me/ap-statistics/unit-1/summary-statistics-for-quantitative-variable/study-guide/fDwLeu9W74iSnEcnKHOA), the Unit 1 overview (https://library.fiveable.me/ap-statistics/unit-1), and practice problems (https://library.fiveable.me/practice/ap-statistics).
I'm confused about percentiles - what does it mean if a score is at the 75th percentile?
If a score is at the 75th percentile it means that 75% of the data are less than or equal to that score and about 25% are higher. In other words, you did better than roughly three-quarters of the people—you’re in the top 25%. This matches the CED definition (UNC-1.I.5). A few practical notes for AP Stats: - The 75th percentile usually equals the third quartile Q3 (UNC-1.I.4), so Q3 marks the upper boundary of the middle 50%. - For a data set of n values, the 75th percentile is near the 0.75(n+1)th ordered value (interpolation may be needed). - Percentiles are measures of position, not spread—use IQR or s for variability (UNC-1.J). Want more practice and examples? Check the Topic 1.7 study guide (https://library.fiveable.me/ap-statistics/unit-1/summary-statistics-for-quantitative-variable/study-guide/fDwLeu9W74iSnEcnKHOA) and try problems at (https://library.fiveable.me/practice/ap-statistics).
How do I know if a data point is an outlier using the 1.5 IQR rule?
Use Q1, Q3, and the interquartile range (IQR = Q3 − Q1). The 1.5×IQR rule says a value is an outlier if it’s either - < Q1 − 1.5×IQR (much smaller than the typical lower half), or - > Q3 + 1.5×IQR (much larger than the typical upper half). How to do it step-by-step: 1. Order the data and find the median, then find Q1 (median of lower half) and Q3 (median of upper half)—those are resistant measures. 2. Compute IQR = Q3 − Q1. 3. Calculate lower fence = Q1 − 1.5×IQR and upper fence = Q3 + 1.5×IQR. 4. Any observation outside those fences is an outlier (you can mark these on a boxplot). This is the method listed in the CED (UNC-1.K.1) and is what AP expects when identifying outliers for Topic 1.7. If you want practice, see the Topic 1.7 study guide (https://library.fiveable.me/ap-statistics/unit-1/summary-statistics-for-quantitative-variable/study-guide/fDwLeu9W74iSnEcnKHOA) and lots of practice problems (https://library.fiveable.me/practice/ap-statistics).
What's the difference between range and interquartile range?
Range = max − min. It’s the simplest measure of spread and uses the two extreme values, so one big outlier can change it a lot. IQR = Q3 − Q1, where Q1 and Q3 are the first and third quartiles (medians of the lower and upper halves). IQR measures the spread of the middle 50% of the data and is resistant (robust) to outliers, while the range and standard deviation are nonresistant. Why that matters: use IQR when the distribution is skewed or has outliers (and when you need to apply the 1.5×IQR rule to flag outliers). Use range for a quick sense of overall spread when extremes matter or the data are fairly symmetric and clean. Both are listed in the CED as common measures of variability (UNC-1.J.1–.2). For AP review, practice computing Q1/Q3, IQR, and the 1.5×IQR outlier rule on the Topic 1.7 study guide (https://library.fiveable.me/ap-statistics/unit-1/summary-statistics-for-quantitative-variable/study-guide/fDwLeu9W74iSnEcnKHOA) and try extra problems at (https://library.fiveable.me/practice/ap-statistics).
When should I use standard deviation vs IQR to measure spread?
Use IQR when the distribution is skewed or has outliers; use standard deviation when the distribution is roughly symmetric with no extreme values. The Course and Exam Description calls mean and standard deviation “nonresistant” (they’re pulled by outliers), while median and IQR are “resistant.” So: - Skewed or outliers present → report median and IQR (IQR = Q3 − Q1; use the 1.5×IQR rule to flag outliers). - Approximately symmetric, no outliers → report mean and standard deviation (s = sqrt[Σ(xi − x̄)²/(n−1)]), and you can use ±2s as a rough outlier guideline. On the AP exam, pick the measure that matches shape and robustness (UNC-1.J, UNC-1.K). For practice, review Topic 1.7 on Fiveable (https://library.fiveable.me/ap-statistics/unit-1/summary-statistics-for-quantitative-variable/study-guide/fDwLeu9W74iSnEcnKHOA) and do practice problems (https://library.fiveable.me/practice/ap-statistics).
Can someone explain step by step how to find the median when I have an even number of data points?
When you have an even number of data points, the median is the average of the two middle values. Step-by-step: 1. Order the data from smallest to largest. 2. Let n be the number of observations (n is even). Find the two middle positions: n/2 and n/2 + 1. 3. Identify the values at those positions in the ordered list (call them m1 and m2). 4. Median = (m1 + m2) / 2. Example: data = {3, 7, 8, 12, 14, 20} (n = 6). Positions n/2 = 3 and n/2+1 = 4 → values 8 and 12. Median = (8 + 12)/2 = 10. AP note (CED UNC-1.I.3): the median can be any value between the two middle values, but AP Stats uses the average of the two middle values as the median for even n. The median is resistant to outliers (useful vs. mean). For more on summary stats, see the Topic 1.7 study guide (https://library.fiveable.me/ap-statistics/unit-1/summary-statistics-for-quantitative-variable/study-guide/fDwLeu9W74iSnEcnKHOA). For tons of practice, check practice problems (https://library.fiveable.me/practice/ap-statistics).
What does it mean for a statistic to be resistant or non-resistant to outliers?
A statistic is resistant (robust) if extreme values (outliers) have little or no effect on it; it’s nonresistant if outliers can change it a lot. For example, the median and IQR are resistant: one very large or small value usually won’t move the median much or change the IQR much. The mean, standard deviation, and range are nonresistant: a single extreme value can pull the mean toward it, inflate s, and blow up the range. AP tip: use resistant measures (median/IQR) when a distribution is skewed or has outliers; use mean/s and range when the distribution is roughly symmetric and outliers aren’t present (UNC-1.K.2). Outliers themselves are commonly flagged by the 1.5×IQR rule or being ±2 standard deviations from the mean (UNC-1.K.1). For a quick review of these ideas and practice, see the Topic 1.7 study guide (https://library.fiveable.me/ap-statistics/unit-1/summary-statistics-for-quantitative-variable/study-guide/fDwLeu9W74iSnEcnKHOA) and hit the AP practice problems (https://library.fiveable.me/practice/ap-statistics).
How do I calculate variance and what's the relationship between variance and standard deviation?
Variance measures the average squared distance of data from the mean. For a sample, the sample variance is s² = (1/(n − 1)) Σ(xi − x̄)². You compute each deviation (xi − x̄), square it, sum those squares, then divide by n − 1. The AP formula sheet gives the sample standard deviation as sx = sqrt[ (1/(n−1)) Σ(xi − x̄)² ], so s = √(s²). That’s the key relationship: standard deviation is the square root of variance. Variance has squared units (e.g., dollars²), while SD is in the original units, so SD is easier to interpret. Because variance/SD use every value and squared deviations, they’re nonresistant (outliers affect them). A common AP rule-of-thumb for outliers: values more than about 2 SDs from the mean may be unusual (CED UNC-1.K.1.ii). For more review and examples see the Topic 1.7 study guide (https://library.fiveable.me/ap-statistics/unit-1/summary-statistics-for-quantitative-variable/study-guide/fDwLeu9W74iSnEcnKHOA) and practice problems (https://library.fiveable.me/practice/ap-statistics).
I don't understand why we use n-1 instead of n in the standard deviation formula - can someone explain?
Good question—AP uses n−1 (Bessel’s correction) because s = sqrt[Σ(xi − x̄)²/(n−1)] gives an unbiased estimate of the population standard deviation’s squared value (the variance). When you compute deviations from the sample mean x̄, you’ve already used the data to estimate the center, so you’ve “lost” one degree of freedom. Dividing by n would systematically underestimate the true population variance on average. Using n−1 corrects that bias so E[s²] ≈ σ². So in AP language: s is a statistic (UNC-1.I.1) that estimates the population SD; the formula with n−1 (UNC-1.J.3) compensates for the fact that the sample mean is itself an estimate. The AP formula sheet gives s with n−1, so use that on the exam (it’s on the provided formula sheet and allowed on calculators). For a quick refresher, see the Topic 1.7 study guide (https://library.fiveable.me/ap-statistics/unit-1/summary-statistics-for-quantitative-variable/study-guide/fDwLeu9W74iSnEcnKHOA) and practice problems (https://library.fiveable.me/practice/ap-statistics).
What's the difference between a statistic and a parameter?
A parameter is a number that describes a population (fixed but usually unknown)—common ones are the population mean μ and population standard deviation σ. A statistic is a number calculated from a sample that summarizes the sample—examples are the sample mean x̄ and sample standard deviation s (CED UNC-1.I.1, UNC-1.I.2, UNC-1.J.3). Key difference: the parameter is what you’d like to know about the whole population; the statistic is what you actually compute from your sample and use to estimate or test that parameter. Statistics vary from sample to sample (sampling variability); parameters don’t. Because the mean and standard deviation are nonresistant, outliers in your sample will affect x̄ and s more than the median or IQR (CED UNC-1.K.2). For more review, see the Topic 1.7 study guide (https://library.fiveable.me/ap-statistics/unit-1/summary-statistics-for-quantitative-variable/study-guide/fDwLeu9W74iSnEcnKHOA), the Unit 1 overview (https://library.fiveable.me/ap-statistics/unit-1), and practice problems (https://library.fiveable.me/practice/ap-statistics).
How do changing units affect my calculated statistics like mean and standard deviation?
Short answer: when you add or multiply every data value by constants, the statistics change in predictable ways. If new value = a·x + b then - Mean: x̄' = a·x̄ + b (shifts and rescales) - Median, percentiles, Q1, Q3: each is a·(old) + b (so adding b shifts them; multiplying by a rescales) - Range and IQR: multiply by |a| (adding b leaves spread unchanged) - Standard deviation: s' = |a|·s (variance multiplies by a^2) - Z-scores: unchanged (they're unitless) Example: convert heights from inches to cm (a = 2.54, b = 0): the mean and s both get multiplied by 2.54. Adding 5 to every score shifts center and percentiles by +5 but leaves SD/IQR the same. This aligns with the CED (UNC-1.I, UNC-1.J; especially UNC-1.J.4). For quick review and examples, check the Topic 1.7 study guide (https://library.fiveable.me/ap-statistics/unit-1/summary-statistics-for-quantitative-variable/study-guide/fDwLeu9W74iSnEcnKHOA). For extra practice, Fiveable has many problems (https://library.fiveable.me/practice/ap-statistics).
If I have outliers in my data set, which measures of center and spread should I report?
If your data have outliers (or are skewed), use resistant summaries: report the median for center and the IQR (Q3 − Q1) for spread. Those won’t be pulled by extreme values the way the mean and standard deviation are (mean and s are nonresistant). You can identify outliers with the 1.5×IQR rule or the ±2 s rule (1.5×IQR beyond Q1 or Q3, or more than 2 s from the mean). If the distribution is roughly symmetric with no outliers, give the mean and standard deviation instead. For AP: always justify your choice of measures (say “median & IQR because distribution is skewed/has outliers”) and show how you identified outliers. For more review and examples, see the Topic 1.7 study guide (https://library.fiveable.me/ap-statistics/unit-1/summary-statistics-for-quantitative-variable/study-guide/fDwLeu9W74iSnEcnKHOA) and Unit 1 overview (https://library.fiveable.me/ap-statistics/unit-1). Practice problems: (https://library.fiveable.me/practice/ap-statistics).