week01-notes.md

CSE206 — Week 01 Notes — Descriptive stats (mean/median/SD, plots)

Lectures: CSE206_Fa24-01.pdf Lab/Tutorial: week01.pdf

1. Big picture (5–10 bullets)

• Probability gives a precise mathematical way to talk about uncertainty in models of the real world.
• Statistics uses data (observations) to learn about the unknown “state of nature” behind those models.
• We describe data sets using descriptive statistics: measures of location (center) and measures of scattering (spread).
• The sample mean, median, and quantiles are different ways to say “where the data sit” on the number line.
• The sample standard deviation and mean absolute deviation describe how tightly or loosely data are clustered.
• Dot plots and histograms are visual tools to see the overall shape of a data set (center, spread, skewness, outliers).
• Outliers are atypical data points that can strongly influence the mean and standard deviation, but much less the median.
• The lab for week 1 gives practice computing these summaries, drawing dot plots and histograms, and comparing two groups.

2. Key concepts and definitions

2.1 Probability vs statistics

• Plain-language definition.
  • Probability is a branch of mathematics that encodes uncertainty and lets us compute chances of events under a model.
  • Statistics uses observed data to infer the unknown parameters or probabilities that define that model.
• Formal definition.
  • In probability, we assume a probability model is known: basic events have assigned probabilities, and we derive the probabilities of more complex events from them.
  • In statistics, we treat those probabilities or parameters as unknown and use data samples to estimate or test them.
• Intuition / mental model.
  • Probability: “If I know how the world works, what is the chance of this outcome?” (e.g., probability of five heads in a row for a fair coin).
  • Statistics: “From what I have observed, what can I say about how the world works?” (e.g., is this coin fair given the toss results?).
• Tiny example.
  • Probability question: assuming a fair coin, how likely is it that heads appears nearly half the time in many tosses?
  • Statistics question: I tossed a coin ten times and got eight heads; can I reasonably claim the coin is not fair?

2.2 State of nature, observations, and samples

• Plain-language definition.
  • The state of nature is the underlying reality or mechanism that generates data.
  • Observations are the actual values we record from that mechanism; the collection of observed values is a sample.
• Formal definition.
  • We assume there is an unknown probability distribution describing the phenomenon.
  • A sample is a finite sequence of real numbers $x_1, \dots, x_n$ drawn from that distribution.
• Intuition / mental model.
  • Think of the state of nature as the complete list of all possible outputs, which we can never fully see, and the sample as a small subset we actually measure.
  • Examples from the lecture: all possible responses of a large AI model, all heights in a city, or all device lifetimes from a factory.
• Tiny example.
  • Population heights: the true heights of everyone in Kazan form the state of nature; the thousand people you measure in a study form your sample.

2.3 Sample mean (measure of location)

• Plain-language definition.
  • The sample mean is the average value of the data; it is one way to describe the center of a sample.
• Formal definition.
  • For a sample ${x_1, \dots, x_n}$, the sample mean is $$ \bar{x} = \frac{1}{n}\sum_{j=1}^n x_j. $$
  • It is the unique number $\mu$ for which $\sum_{j=1}^n (x_j - \mu) = 0$.
• Intuition / mental model.
  • Imagine the data points as weights on a number line; the mean is the balance point where the system is in equilibrium.
  • It is very sensitive to extreme values (outliers), because every data point enters the sum directly.
• Tiny example.
  • If reaction times in a tiny sample are $180, 220, 200$ milliseconds, then $\bar{x} = (180 + 220 + 200)/3 = 200$ ms.

2.4 Median and quantiles (more location measures)

• Plain-language definition.
  • The median is the “middle” value of an ordered sample.
  • Quantiles (such as quartiles) mark positions that split the ordered data into fixed fractions (like quarters).
• Formal definition.
  • Reorder the sample into an increasing sequence $y_1 \le \dots \le y_n$.
  • If $n$ is odd, the median $m$ is $y_{(n+1)/2}$.
  • If $n$ is even, the median is $$ m = \frac{y_{n/2} + y_{n/2 + 1}}{2}. $$
  • For a positive integer $q$, the $k$-th $q$-quantile is the value $y_j$ such that roughly $k/q$ of the values lie below $y_j$ and the rest above it.
• Intuition / mental model.
  • The median is a center that is robust: moving a few extreme points far away does not change it much, as long as the order around the middle stays the same.
  • Quartiles cut the ordered data into four blocks of equal size and help describe spread and skewness.
• Tiny example.
  • For the ordered data ${1, 3, 5, 9, 10}$ ($n = 5$), the median is the third value: $m = 5$.
  • For ${1, 3, 5, 9}$ ($n = 4$), the median is $(3 + 5)/2 = 4$.

2.5 Standard deviation and mean absolute deviation (measures of scattering)

• Plain-language definition.
  • The standard deviation measures how far, on average in a squared sense, data points are from the mean.
  • The mean absolute deviation (m.a.d.) measures how far, on average, data points are from the mean using absolute distances.
• Formal definition.
  • For a sample ${x_1, \dots, x_n}$ with mean $\bar{x}$, the (sample) standard deviation is $$ s = \sqrt{\frac{1}{n - 1}\sum_{j=1}^n (x_j - \bar{x})^2}. $$

    The denominator $n-1$ is the “degrees of freedom” correction (Bessel’s correction): we already used the data once to estimate $\bar{x}$, so if we divided by $n$ the average squared deviation would systematically underestimate the true variance of the population. With $1/(n-1)$, $s^2$ is (under standard i.i.d. assumptions) an unbiased estimator of the population variance.

  • The sample variance ($\operatorname{Var}(x)$) is $s^2$, the square of the standard deviation.
  • If instead you are treating the data set as the whole population (for example, measuring spread of all exam scores in a fixed class, or computing loss over all points in a finite training set), it is natural to drop the correction and define the population variance by dividing by $n$.
  • The mean absolute deviation is $$ \text{m.a.d.} = \frac{1}{n}\sum_{j=1}^n |x_j - \bar{x}|. $$
• Intuition / mental model.
  • Both numbers describe how “together” or “spread out” the data are, but standard deviation penalizes large deviations more strongly because of the square.
  • Standard deviation and m.a.d. are insensitive to adding a constant to all data (a shift), but they scale when all data are multiplied by a constant.
• Tiny example.
  • If a small sample is ${9, 10, 11}$, then $\bar{x} = 10$.
  • The deviations are $-1, 0, +1$; the m.a.d. is $(1 + 0 + 1)/3 = 2/3$.
  • The standard deviation is $\sqrt{[(1^2 + 0^2 + 1^2)/(3 - 1)]} = \sqrt{1} = 1$.

2.6 Dot plots and histograms (data representation)

• Plain-language definition.
  • A dot plot places a dot for each data point in one of several equal-width cells along the horizontal axis.
  • A histogram uses rectangles over those cells; the height of each rectangle is the frequency (or relative frequency) of data in that cell.
• Formal definition.
  • Let $x_{\min}$ and $x_{\max}$ be the minimum and maximum data values.
  • Choose a number of cells $N$ and partition $[x_{\min}, x_{\max}]$ into $N$ subintervals of equal length.
  • Dot plot: for each subinterval, draw a column of dots, one per data point in that interval.
  • Histogram: for each subinterval, draw a rectangle with base the subinterval and height equal to the number of data points (or relative frequency) in that interval.
• Intuition / mental model.
  • Dot plots are very clear for small to moderate samples: you can see each point and where it falls.
  • Histograms are better for large samples, letting you see overall shape (e.g., symmetric, skewed, multimodal) without tracking individual values.
• Tiny example.
  • For the sample ${10, -2, -4.5, -7.8, -1, 0, 0, 0, 3, 3, 0, 5, -1, 0, -2, -2, -7.7, 10, 10, 11, -2, -3}$ from the lecture, the instructor shows dot plots and histograms that change shape when the number of cells $N$ changes.

2.7 Outliers and empirical distribution function

• Plain-language definition.
  • Outliers are data points that look atypical or far from the bulk of the sample.
  • The empirical distribution function (EDF), denoted $F_{\text{emp}}(x)$, describes, for each $x$, the proportion of data values that are $\le x$.
• Formal definition.
  • For a sample ${x_1, \dots, x_n}$, the EDF is $$ F_{\text{emp}}(x) = \frac{1}{n}#{ j \in {1, \dots, n} : x_j \le x}. $$
  • Outliers are not defined by a single formula in the lecture; their identification depends on the underlying state of nature and context.
• Intuition / mental model.
  • $F_{\text{emp}}(x)$ is a step function that jumps up by $1/n$ at each data point; it shows how the sample “accumulates” as $x$ increases.
  • Outliers may lie outside a range such as $(\bar{x} - s, \bar{x} + s)$, but in many real problems this simple rule can mislead, so context is crucial.
• Tiny example.
  • For the sample ${22, 24, 35, 1, 1, 5, 6, 10, 13, 21, 10, 10, 40, 41, 47}$ from the lecture, $F_{\text{emp}}(x)$ is $0$ for $x < 1$, jumps to $2/15$ at $x = 1$, and eventually reaches $1$ after the largest value $47$.

3. Core formulas and how to use them

3.1 Sample mean

• Formula.
  • $$ \bar{x} = \frac{1}{n}\sum_{j=1}^n x_j. $$
• Symbols.
  • $x_1, \dots, x_n$: observed data values (the sample).
  • $n$: sample size.
  • $\bar{x}$: sample mean (average).
• When to use it.
  • To summarize the central tendency of quantitative data when you are not too worried about extreme values.
  • In many later results in probability and statistics, the sample mean is a basic building block.
• Common mistakes.
  • Forgetting to divide by $n$ after summing.
  • Using rounded intermediate sums that cause unnecessary rounding error.
  • Treating the mean as robust to outliers (it is not).

3.2 Median and quantiles

• Formula / definition.
  • Order the data: $y_1 \le \dots \le y_n$.
  • Median $m$:
    • If $n$ is odd, $m = y_{(n+1)/2}$.
    • If $n$ is even, $m = (y_{n/2} + y_{n/2+1})/2$.
  • First and third quartiles are special quantiles located at roughly 25% and 75% positions in the ordered list.
• Symbols.
  • $y_j$: $j$-th ordered value of the sample.
  • $n$: sample size.
  • $m$: sample median.
• When to use it.
  • To describe the center of data sets with clear skewness or outliers, where the mean might be misleading.
  • To split the data into parts and reason about where most values lie.
• Common mistakes.
  • Forgetting to order the data before taking the middle.
  • Miscounting indices when $n$ is even.
  • Confusing median (position-based) with mean (value-based).

3.3 Trimmed mean (used in the lab)

• Formula / definition.
  • For a $p%$ trimmed mean:
    1. Order the data.
    2. Remove the smallest $p%$ and largest $p%$ of observations.
    3. Take the ordinary mean of the remaining data.
  • The lab uses a $10%$ trimmed mean.
• Symbols.
  • $p$: trimming percentage.
  • $n$: sample size; $pn$ (rounded appropriately) observations are trimmed from each tail.
• When to use it.
  • When you want a measure of center that behaves like the mean but is less sensitive to a few extreme values.
  • To check whether outliers are heavily influencing the plain mean by comparing mean vs trimmed mean vs median.
• Common mistakes.
  • Forgetting to sort the data before trimming.
  • Removing the wrong number of observations from each tail.
  • Mixing up which data remain after trimming when computing the average.

3.4 Standard deviation and variance

• Formula.
  • Standard deviation: $$ s = \sqrt{\frac{1}{n - 1}\sum_{j=1}^n (x_j - \bar{x})^2}. $$
  • Variance: $$ s^2 = \frac{1}{n - 1}\sum_{j=1}^n (x_j - \bar{x})^2. $$
• Symbols.
  • $x_j$: data points.
  • $\bar{x}$: sample mean.
  • $n$: sample size.
  • $s$: sample standard deviation.
  • $s^2$: sample variance.
• When to use it.
  • To measure spread around the mean and compare variability between samples (e.g., control vs treatment groups in the lab).
  • As input to many later statistical procedures (confidence intervals, tests, etc.).
• Common mistakes.
  • Forgetting the context of the denominator:
    • use $n-1$ when the data are a sample from a larger population and you want $s^2$ to estimate the unknown population variance (unbiased “sample” variance);
    • use $n$ when you are describing the spread of a finite population you have completely observed (population variance).
  • Forgetting to square the deviations before summing.
  • Taking the square root too early or forgetting it when going from variance to standard deviation.

3.5 Mean absolute deviation (m.a.d.)

• Formula.
  • $$ \text{m.a.d.} = \frac{1}{n}\sum_{j=1}^n |x_j - \bar{x}|. $$
• Symbols.
  • $x_j$: data points.
  • $\bar{x}$: sample mean.
  • $n$: sample size.
  • $|\cdot|$: absolute value.
• When to use it.
  • When you want a measure of spread that treats all deviations linearly instead of squaring them.
  • As a way to see how much data typically differ from the mean without giving extra emphasis to outliers.
• Common mistakes.
  • Confusing m.a.d. with standard deviation; they are different quantities.
  • Forgetting to take absolute values before summing.

3.6 Empirical distribution function (EDF)

• Formula.
  • $$ F_{\text{emp}}(x) = \frac{1}{n}#{ j \in {1, \dots, n} : x_j \le x}. $$
  • In words: the EDF takes your sample values $x_1,\dots,x_n$ and, for any cutoff $x$ you choose, gives back the fraction of sample points that are $\le x$. If you plot $x\mapsto F_{\text{emp}}(x)$, you get a step graph that shows how the data accumulate from left to right.
• Symbols.
  • $x$: point on the horizontal axis where we evaluate the function.
  • $x_j$: data points.
  • $n$: sample size.
  • $#{\cdot}$: number of elements in the set.
• When to use it.
  • To visualize how the sample accumulates as values increase.
  • To connect sample behavior with quantiles (e.g., where $F_{\text{emp}}$ crosses 0.25, 0.5, 0.75).
• Common mistakes.
  • Counting only strictly less than instead of “less than or equal”, which changes the step locations.
  • Forgetting to divide by $n$, giving counts instead of proportions.

4. Worked examples

4.1 Effect of a single extreme value on mean, median, standard deviation, and m.a.d.

• Setup.
  • From the lecture: suppose we have one thousand data points, one of which is $1000$ and all the rest are $0$.
  • So the sample is $x_1, \dots, x_{1000}$ with 999 zeros and a single 1000.
• Step 1: compute the sample mean.
  • The sum of all values is $1000 + 0 + \dots + 0 = 1000$.
  • The sample size is $n = 1000$.
  • The mean is $$ \bar{x} = \frac{1000}{1000} = 1. $$
• Step 2: compute the median.
  • Order the data: 999 zeros, then one 1000 at the end.
  • For $n = 1000$, the median is the average of the 500th and 501st ordered values.
  • Both the 500th and 501st values are 0, so the median is $$ m = \frac{0 + 0}{2} = 0. $$
• Step 3: compute the standard deviation.
  • Deviations from the mean: 999 values equal to $(0 - 1) = -1$, and one value equal to $(1000 - 1) = 999$.
  • Sum of squared deviations: $$ \sum_{j=1}^{1000} (x_j - \bar{x})^2 = 999^2 + 999 \cdot 1^2 = 999 \cdot 1000 = 999000. $$
  • The standard deviation is $$ s = \sqrt{\frac{999000}{1000 - 1}} = \sqrt{\frac{999000}{999}} = \sqrt{1000} \approx 31.6. $$
• Step 4: compute the mean absolute deviation.
  • Absolute deviations: $|1000 - 1| = 999$, and 999 copies of $|0 - 1| = 1$.
  • Sum of absolute deviations is $999 + 999 \cdot 1 = 1998$.
  • m.a.d. is $$ \text{m.a.d.} = \frac{1998}{1000} = 1.998. $$
• Check your intuition.
  • The mean (1) lies close to the many zeros but is pulled away by the single large value.
  • The median (0) completely ignores the extreme 1000.
  • The standard deviation is very large because the squared deviation of 999 dominates.
  • The m.a.d. is closer in size to the mean and much smaller than the standard deviation.

4.2 Empirical distribution function and quantiles

• Setup.
  • From the lecture, consider the sample ${22, 24, 35, 1, 1, 5, 6, 10, 13, 21, 10, 10, 40, 41, 47}$.
  • There are $n = 15$ observations.
• Step 1: order the data.
  • Ordered sample: ${1, 1, 5, 6, 10, 10, 10, 13, 21, 22, 24, 35, 40, 41, 47}$.
• Step 2: compute $F_{\text{emp}}(x)$ at a few points.
  • For $x < 1$, no observations are $\le x$, so $F_{\text{emp}}(x) = 0$.
  • At $x = 1$, two observations are $\le 1$, so $$ F_{\text{emp}}(1) = \frac{2}{15}. $$
  • At $x = 10$, seven observations are $\le 10$, so $$ F_{\text{emp}}(10) = \frac{7}{15}. $$
  • For any $x$ between 41 and 47 (but less than 47), 14 observations are $\le x$, so $F_{\text{emp}}(x) = 14/15$.
  • For $x \ge 47$, all 15 observations are $\le x$, so $F_{\text{emp}}(x) = 1$.
• Step 3: relate to quantiles.
  • The median is the 8th ordered value (since $n = 15$), which is $13$.
  • At $x = 13$, exactly 8 observations are $\le 13$, so $$ F_{\text{emp}}(13) = \frac{8}{15} \approx 0.53. $$
  • This is just above 0.5, consistent with the interpretation of the median as the 50%-quantile.
• Check your intuition.
  • As $x$ moves along the horizontal axis, the EDF jumps up by $1/15$ at each data point.
  • Reading off approximate levels like 0.25, 0.5, and 0.75 on the vertical axis helps locate quartiles and median.

5. Lab/Tutorial essentials (week01.pdf)

5.1 What the lab asked you to do

• Problem 1 (Walpole 1.2): reaction time of 20 amateur racers.
  • Compute the sample mean and sample median of the reaction-time data.
  • Compute the 10% trimmed mean.
  • Draw a dot plot of the data.
  • Compare mean, median, and trimmed mean to discuss possible outliers.
• Problem 2 (Walpole 1.5 and 1.11): effect of regular exercise on cholesterol.
  • Data are changes in cholesterol levels for a control and a treatment group.
  • Draw a dot plot for both groups on the same graph.
  • Compute mean, median, and 10% trimmed mean for each group.
  • Explain why the mean difference suggests one conclusion while median/trimmed mean suggest another.
  • Compute sample variance and sample standard deviation for each group.
• Problem 3 (Walpole 1.13): battery lifetime data (hours).
  • Compute sample mean and median.
  • Identify which value causes a substantial difference between mean and median.
  • Recalculate the mean without that value and compare.
• Problem 4 (Walpole 1.17): time to fall asleep for smokers (A) and nonsmokers (B).
  • Compute sample mean and sample standard deviation for each group.
  • Draw a dot plot for both groups on the same line.
  • Comment on how smoking appears to affect time to fall asleep.
• Problem 5 (Walpole 1.18): final exam scores in an elementary statistics course.
  • Construct a relative frequency histogram.
  • Sketch a smooth curve that approximates the distribution and discuss skewness.
  • Compute sample mean, sample median, and sample standard deviation.
• Problem 6 (Walpole 1.20): lifetime (seconds) of 50 fruit flies under a new spray.
  • Set up a relative frequency distribution.
  • Construct a relative frequency histogram.
  • Find the median.
• Problem 7: complete the Colab notebook linked at the end of the lab (practice using Python/Colab for descriptive statistics).

5.2 How to solve / approach them

• Computing means, medians, trimmed means.
  • Sum all values and divide by the sample size to get the mean.
  • Sort the data; take the middle value(s) to find the median.
  • For a 10% trimmed mean, sort the data, remove the smallest and largest 10% of observations, then compute the mean of what remains.
• Dot plots and histograms.
  • For dot plots, choose a horizontal axis that covers all data, split into equal-width cells, and place a dot in the appropriate cell for each observation.
  • For histograms, count how many observations fall in each cell (frequency) or divide by the total number of observations (relative frequency), then draw rectangles with those heights.
  • When comparing two groups (e.g., control vs treatment, smokers vs nonsmokers), plot them on the same axis to compare centers, spreads, and outliers visually.
• Interpreting outliers and robustness.
  • If the mean, median, and trimmed mean are close, there is little evidence of strong outliers.
  • If the mean is far away from the median and trimmed mean, this suggests that a few extreme values may be pulling the mean.
  • Recomputing the mean after removing a suspected extreme value is a way (used in the lab) to see how influential it is.
• Comparing groups with descriptive statistics.
  • For cholesterol and sleep-time problems, compare means to see differences in average level between groups.
  • Compare medians and trimmed means to see whether group differences are robust to outliers.
  • Compare standard deviations to see which group has more variability.
  • When commenting on “impact” (e.g., of smoking or treatment), base your reasoning on these differences and on the plots (dot plots, histograms).
• Relative frequency distributions and skewness.
  • To build a relative frequency distribution, define class intervals (bins), count how many observations fall in each, and divide by the total number of observations.
  • Plot these relative frequencies as a histogram and inspect whether the distribution is symmetric, right-skewed (long tail to the right), or left-skewed.
  • Use the shape to interpret how the bulk of the data and extreme values behave.

5.3 Mini practice

• Practice 1: mean vs median vs outlier.
  • Question: Consider a data set of 9 exam scores where 8 scores are between 60 and 80, and one score is 10. Which measure of center (mean or median) is more affected by the single low score?
  • Brief answer: The mean is much more affected, because it averages all values; the median depends only on the order and usually stays near the middle of the bulk of the data.
• Practice 2: interpreting a dot plot of two groups.
  • Question: In a dot plot, group A’s dots are mostly clustered tightly around 0 with a few large positive values, while group B’s dots are spread more evenly from negative to positive values. Which group likely has larger standard deviation, and what might that mean?
  • Brief answer: Group B likely has a larger standard deviation, indicating more variability in its measurements. Group A has a tighter core but a few outliers.
• Practice 3: trimmed mean reasoning.
  • Question: You compute a mean of 20 for a sample, but the 10% trimmed mean and the median are both close to 15. What does this suggest about the data?
  • Brief answer: It suggests there are some extreme high values that pull the plain mean upward; trimming and using the median reduce their influence and give a center near 15.

6. Quick recap

• Probability assumes a model with known probabilities and computes chances of events; statistics uses data to infer those unknown probabilities or parameters.
• The state of nature is the underlying reality (or probability distribution); a sample is the finite list of observed values.
• The sample mean $\bar{x} = \frac{1}{n}\sum x_j$ is a natural measure of center but is sensitive to outliers.
• The median and other quantiles are location measures based on ordered positions and are more robust to extreme values.
• The sample standard deviation $s$ and variance $s^2$ measure how spread out the data are around the mean, with large deviations heavily weighted.
• The mean absolute deviation measures spread using absolute distances and usually gives a smaller value than the standard deviation for the same data.
• Dot plots and histograms turn raw data into visual summaries that reveal center, spread, skewness, and potential outliers.
• Outliers are atypical points whose importance must be judged in context; simple rules based on $(\bar{x} - s, \bar{x} + s)$ can be misleading.
• The empirical distribution function $F_{\text{emp}}(x)$ counts the proportion of observations $\le x$ and connects the sample to quantiles like the median.
• The week 1 lab reinforces these ideas by having you compute means, medians, trimmed means, standard deviations, draw dot plots and histograms, and interpret differences between groups.