• Individual data points can jump around from each other, the averages smooth these flucuations
• The sample mean: \[ \bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i \] where each \(X_i\) is one observation
• As the sample size \(n\) increases, variability of \(\bar{X}\) shrinks.
• The mean \(\bar{X}\) “homes in” on the true population mean \(\mu\).
• The collection of all possible sample means forms the
  sampling distribution of the mean,
  which shows how the mean varies from sample to sample.

• Repeated samples of size \(n\) make the sample mean \(\bar{X}\) vary less than individual data.
  
• Variance of the mean:
  \[ Var(\bar{X}) = \frac{\sigma^2}{n} \]
• Standard error (spread of sample means):
  \[ SE = \frac{\sigma}{\sqrt{n}} \]
• As \(n\) increases, \(SE\) decreases — larger samples give more precise estimates.

• To see how sample size affects stability, we’ll simulate data from a skewed population.
  
• Use an Exponential distribution (mean = 1, variance = 1).
  
• For each sample size \(n \in \{5, 30, 100\}\):
  • Draw many random samples.
  • Compute each sample’s mean \(\bar{X}\).
• Compare how the spread of sample means changes as \(n\) increases.

## # A tibble: 3 × 3
##       n emp_mean emp_sd
##   <dbl>    <dbl>  <dbl>
## 1     5    1.00  0.445 
## 2    30    1.00  0.185 
## 3   100    0.999 0.0988

• Each curve shows the distribution of sample means \(\bar{X}\) for a different sample size.
  
• Population: Exponential(1) (skewed, mean = 1).
  
• Sample sizes \(n = 5, 30, 100\).
  
• As \(n\) increases:
  • The curves narrow (less variability).
  • The shapes become more symmetric and bell-shaped.
• The dashed line marks the true mean \(\mu = 1\).

    dens_df <- sample_means |>
        group_by(n) |>
        reframe({
            d <- density(mean)
            tibble(x = d$x, y =d$y)
        })
    
    ggplot(dens_df, aes(x, y, color = factor(n), group = n)) +
        geom_line(linewidth = 1.1) +
        geom_vline(xintercept = 1, linetype = 3, linewidth = 0.7) +
        scale_color_manual(values = c("5"="#5B8FD9","30"="#58B88A","100"="#C76C5B"),
                     name = "Sample size n") +
        labs(title = "Sample Means Tighten as n Increases",
           subtitle = "Population: Exponential(1); dashed = true mean (μ = 1)",
           x = expression(bar(X)), y = "Density") +
      theme_minimal(base_size = 14) +
      theme(legend.position = "top", panel.grid.minor = element_blank())

• Each box shows the distribution of sample means \(\bar{X}\) for a given sample size.
  
• Population: Exponential(1) (mean = 1).
  
• As \(n\) increases:
  • The boxes get narrower, showing less spread.
  • The medians stay close to the true mean \(\mu = 1\).
• The dashed line marks the true population mean.
• Bigger samples → more precise estimates of the mean.

• This 3D plot shows how the distribution of sample means changes with sample size.
  
• Axes:
  • x: sample size \(n\)
  • y: sample mean \(\bar{X}\)
  • z: estimated density (frequency)
• As \(n\) increases:
  • The ridge narrows and steepens near the true mean \(\mu = 1\).
  • The density becomes more concentrated — the means vary less.
• Visually confirms that larger n → less spread in the sampling distribution.

## Warning: 'layout' objects don't have these attributes: 'scence'
## Valid attributes include:
## '_deprecated', 'activeshape', 'annotations', 'autosize', 'autotypenumbers', 'calendar', 'clickmode', 'coloraxis', 'colorscale', 'colorway', 'computed', 'datarevision', 'dragmode', 'editrevision', 'editType', 'font', 'geo', 'grid', 'height', 'hidesources', 'hoverdistance', 'hoverlabel', 'hovermode', 'images', 'legend', 'mapbox', 'margin', 'meta', 'metasrc', 'modebar', 'newshape', 'paper_bgcolor', 'plot_bgcolor', 'polar', 'scene', 'selectdirection', 'selectionrevision', 'separators', 'shapes', 'showlegend', 'sliders', 'smith', 'spikedistance', 'template', 'ternary', 'title', 'transition', 'uirevision', 'uniformtext', 'updatemenus', 'width', 'xaxis', 'yaxis', 'boxmode', 'barmode', 'bargap', 'mapType'

• The sample mean \(\bar{X}\) becomes more stable as \(n\) increases.
  
• Variability of the mean:
  \[ Var(\bar{X}) = \frac{\sigma^2}{n} \]
• Standard error (spread of sample means):
  \[ SE = \frac{\sigma}{\sqrt{n}} \]
• Bigger \(n\) → smaller \(SE\) → more precise estimates.
  
• This relationship drives confidence intervals, hypothesis tests, and the CLT.