What is the difference between a Confidence Interval and a Prediction Interval?

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Two types of intervals that are often used in regression analysis are confidence intervals and prediction intervals.

Here’s the difference between the two intervals:

Confidence intervals represent a range of values that are likely to contain the true mean value of some response variable based on specific values of one or more predictor variables.

Prediction intervals represent a range of values that are likely to contain the true value of some response variable for a single new observation based on specific values of one or more predictor variables.

For example, suppose we fit a that uses the number of bedrooms to predict the selling price of a house:

Price = β₀ + β₁(number of bedrooms)

If we’d like to estimate the mean selling price of houses with three bedrooms, we would use a confidence interval.

However, if we’d like to estimate the selling price of a specific new home that just came on the market with three bedrooms, we would use a prediction interval.

Note: Since prediction intervals attempt to create an interval for a specific new observation, there’s more uncertainty in our estimate and thus prediction intervals are always wider than confidence intervals.

Confidence Interval vs. Prediction Interval: Difference in Formulas

We use the following formula to calculate a confidence interval:

ŷ₀  +/-  t_α/2,n-2 * S_yx√((x₀ – x̄)²/SS_x + 1/n)

We use the following formula to calculate a prediction interval:

ŷ₀  +/-  t_α/2,n-2 * S_yx√((x₀ – x̄)²/SS_x + 1/n + 1)

where:

• ŷ₀: Estimated mean value of response variable
• t_α/2,n-2: t-critical value with n-2 degrees of freedom
• S_yx: Standard error of response variable
• x₀: specific value of predictor variable 
• x̄: mean value of predictor variable
• SS_x: Sum of squares for predictor variable
• n: Total sample size

Notice that the formula for a prediction interval contains an extra one in the square root portion, which means the standard error will always be larger than a confidence interval.

Example: Interpreting Confidence Intervals vs. Prediction Intervals

Suppose we have the following dataset that shows the number of bedrooms and the selling price for 20 houses in a particular neighborhood:

Now suppose we fit a simple linear regression model to this dataset in R:

#define data
df <- data.frame(beds=c(1, 1, 1, 2, 2, 2, 2, 3, 3, 3,
                        3, 3, 3, 3, 4, 4, 4, 5, 5, 6),
                 price=c(120, 133, 139, 185, 148, 160, 192, 205, 244, 213,
                         236, 280, 275, 273, 312, 311, 304, 415, 396, 488))

#fit simple linear regression model
model <- lm(price~beds, data=df)

#view model fit
summary(model)

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   39.450     13.248   2.978  0.00807 ** 
beds          70.667      4.031  17.529 9.26e-13 ***
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Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 24.19 on 18 degrees of freedom
Multiple R-squared:  0.9447,	Adjusted R-squared:  0.9416 
F-statistic: 307.3 on 1 and 18 DF,  p-value: 9.257e-13

The fitted regression model turns out to be:

Selling price (thousands) = 39.450 + 70.667(number of bedrooms)

We can use the following code to calculate a confidence interval for the mean selling price of houses that have three bedrooms:

#define new house
new <- data.frame(beds=c(3))

#confidence interval for mean selling price of house with 3 bedrooms
predict(model, newdata = new, interval = "confidence")

     fit     lwr     upr
1 251.45 240.087 262.813

The 95% confidence interval for the mean selling price of a house with three bedrooms is [$240k, $262k].

We can then use the following code to calculate a prediction interval for the selling price of a new house that just came on the market that has three bedrooms:

#define new house
new <- data.frame(beds=c(3))

#confidence interval for mean selling price of house with 3 bedrooms
predict(model, newdata = new, interval = "prediction")

     fit      lwr      upr
1 251.45 199.3783 303.5217

The 95% prediction interval for the selling price of a new house with three bedrooms is [$199k, $303k].

Notice that the prediction interval is much wider than the confidence interval because there is more uncertainty around the selling price of a single new house as opposed to the mean selling price of all houses with three bedrooms.

The following tutorials offer additional information about confidence intervals:

The following tutorials offer additional information about prediction intervals: