1. Descriptive statistics & boxplot
Click on the strip to add data points (or randomise). The five-number summary, mean, variance and standard deviation update live. Mean $\bar x=\frac1n\sum x_i$; sample variance $s^2=\frac1{n-1}\sum (x_i-\bar x)^2$.
Click empty space to add a value; click near a dot to remove it.
2. Mean vs median — outlier sensitivity
Drag the outlier slider and watch the mean chase it while the median stays put. The mean is pulled by extreme values; the median is a resistant measure of centre.
3. Histogram, dotplot & skew
A sample drawn from a skewable population, binned into a histogram with overlaid mean and median. Skew shifts the mean toward the long tail. Adjust skew, sample size and bin count.
4. Conditional probability & independence
A unit square split by two events $A$ and $B$. Drag the partitions to set $P(A)$, $P(B\mid A)$ and $P(B\mid A^c)$. Read $P(A\cap B)$, $P(B)$ and $P(A\mid B)$ via Bayes; events are independent iff $P(B\mid A)=P(B\mid A^c)$.
5. Counting — permutations vs combinations
Choosing $k$ from $n$: order-sensitive permutations $P(n,k)=\frac{n!}{(n-k)!}$ versus unordered combinations $C(n,k)=\binom{n}{k}=\frac{n!}{k!\,(n-k)!}$. The bars compare the two counts.
6. Binomial distribution
$X\sim\mathrm{Bin}(n,p)$ counts successes in $n$ independent Bernoulli trials. PMF $P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}$, mean $np$, variance $np(1-p)$.
7. Poisson distribution
$X\sim\mathrm{Pois}(\lambda)$ models rare-event counts. PMF $P(X=k)=\dfrac{e^{-\lambda}\lambda^k}{k!}$, with mean $=$ variance $=\lambda$. As $n\to\infty$, $p\to0$ with $np=\lambda$, the binomial converges to this.
8. Normal distribution & the 68–95–99.7 rule
$X\sim N(\mu,\sigma^2)$ with density $f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/2\sigma^2}$. Shade $[a,b]$ to read $P(a\le X\le b)$ and the $z$-scores. The empirical rule covers $\pm1,\pm2,\pm3\sigma$.
9. Continuous distributions — PDF & CDF
Uniform, exponential and normal densities side by side with their cumulative distribution $F(x)=P(X\le x)$. Move the cursor line to read the density and the cumulative probability at a point.
10. Covariance, correlation & least squares
Drag points to reshape the cloud. The least-squares line $\hat y=a+bx$ minimises squared residuals; $r=\frac{\mathrm{Cov}(x,y)}{s_x s_y}$ measures linear association and $R^2=r^2$ the variance explained.
Drag any point; click empty space to add one.
11. Central Limit Theorem — sampling distribution of $\bar X$
Repeatedly draw samples of size $n$ from a non-normal population and histogram their means. The sampling distribution of $\bar X$ tends to $N\!\left(\mu,\sigma^2/n\right)$ as $n$ grows, whatever the population.
12. Law of Large Numbers
Flip a biased coin repeatedly; the running proportion of heads converges to the true $p$. Short runs wander; long runs settle. This is the empirical face of probability.