2026-11-01
Point Estimate: a value used to approximate a parameter of a population.
In statistics, this estimation comes from the sample. For instance, the sample mean would be the point estimate, or best guess, of the population mean. Here are a couple of examples:
-The sample mean \(\bar{x}\) is the point estimate for \(\mu\)
-The sample proportion \(\hat{p}\) is the point estimate of p
-The sample variance \(s^{2}\) is the point estimate of \(\sigma^{2}\)
An important use of point estimates is within confidence intervals.
Confidence Interval: a range of values that the true estimate is expected to fall between if the experiment is repeated various times
Example: formula for confidence interval for the population mean (\(\mu\)) with a known population standard deviation (\(\sigma\))
\(\text{Confidence Interval} = \bar{x} \pm z_{\alpha/2}(\frac{\sigma}{\sqrt{n}})\)
with \(\bar{x}\) being the sample mean and point estimate, \(z_{\alpha/2}\) being the critical value, \(\frac{\sigma}{\sqrt{n}}\) and being the standard error
iris Dataset# Snapshot of the first five rows of the iris dataset head(iris,10)
Sepal.Length Sepal.Width Petal.Length Petal.Width Species 1 5.1 3.5 1.4 0.2 setosa 2 4.9 3.0 1.4 0.2 setosa 3 4.7 3.2 1.3 0.2 setosa 4 4.6 3.1 1.5 0.2 setosa 5 5.0 3.6 1.4 0.2 setosa 6 5.4 3.9 1.7 0.4 setosa 7 4.6 3.4 1.4 0.3 setosa 8 5.0 3.4 1.5 0.2 setosa 9 4.4 2.9 1.4 0.2 setosa 10 4.9 3.1 1.5 0.1 setosa
iris Dataset (con’t)# Internal structure of iris dataset str(iris)
'data.frame': 150 obs. of 5 variables: $ Sepal.Length: num 5.1 4.9 4.7 4.6 5 5.4 4.6 5 4.4 4.9 ... $ Sepal.Width : num 3.5 3 3.2 3.1 3.6 3.9 3.4 3.4 2.9 3.1 ... $ Petal.Length: num 1.4 1.4 1.3 1.5 1.4 1.7 1.4 1.5 1.4 1.5 ... $ Petal.Width : num 0.2 0.2 0.2 0.2 0.2 0.4 0.3 0.2 0.2 0.1 ... $ Species : Factor w/ 3 levels "setosa","versicolor",..: 1 1 1 1 1 1 1 1 1 1 ...
This is the dataset I will be working with for the remainder of the presentation. With this, I will demonstrate how to visualize the point estimate and confidence interval.
This is my plotly code of Sepal Length vs. Petal Length of the iris dataset, a line of best fit, and a visualization of a 95% confidence interval
# Linear regression model
mod <- lm(Petal.Length ~ Sepal.Length, data = iris)
# x axis
xax <- list(title = "Sepal Length")
# y axis
yax <- list(title = "Petal Length")
# 95% confidence intervals
preds <- predict(mod, newdata = data.frame(Sepal.Length = iris$Sepal.Length), interval = "confidence")
# Plotly plot of scatter, regression model, and confidence interval
plot_ly(iris,
x = ~Sepal.Length,
y = ~Petal.Length,
type = "scatter",
mode = "markers",
name = "data",
height = 450,
width = 700) %>%
add_lines(x = iris$Sepal.Length, fitted(mod),
name = "Overall Fit",
inherit = FALSE) %>%
add_ribbons(x = ~Sepal.Length,
ymin = preds[ , "lwr"],
ymax = preds[ , "upr"],
name = "95% Confidence Interval",
inherit = FALSE,
line = list(color = 'transparent'),
fillcolor = 'rgba(0, 0, 0, 0.15)',
showlegend = TRUE) %>%
layout(xaxis = xax, yaxis = yax) %>%
config(displayModeBar = FALSE) %>%
onRender('
function(el, x) {
setTimeout(function() {
window.dispatchEvent(new Event("resize"));
}, 250);
}
')
This is the resulting interactive graph with the confidence interval.
When the data of sepal width is arranged in a histogram, it is possible to estimate the point estimate for mean sepal width. It’d be more exact to calculate the average of the sample, but you can tell from first glance that the mean is around 3.0.
This is a visualization of both the point estimate and the confidence interval for the means of each iris species in the iris datset. 