Understanding Percentiles
Percentiles are values below which a certain percentage of data in a dataset falls. For example, the 25th percentile (first quartile) is the value below which 25% of the data falls. Given three percentiles, we can approximate the underlying distribution of the data.
Steps to Estimate the Distribution
Now we discuss step by step the Estimate of the Distribution for Three Percentiles in R Programming Language.
Step 1: Visual Inspection
First, let’s visualize these percentiles on a plot to understand their spread.
# Define the percentiles
percentiles <- c(5, 15, 25)
names(percentiles) <- c("P10", "P50", "P90")
# Plot the percentiles
plot(1:3, percentiles, xaxt = "n", ylab = "Value", xlab = "Percentile",
main = "Percentile Plot")
axis(1, at = 1:3, labels = names(percentiles))
Output:
Estimate a Distribution Based on Three Percentiles in R
The plot will visually represent the specified percentiles as follows:
- The x-axis will show the labels P10, P50, and P90.
- The y-axis will show the values 5, 15, and 25 corresponding to these percentiles.
- The plot will have points at (1, 5), (2, 15), and (3, 25) connected by lines since the default type “b” is used in the plot function.
Step 2: Fit a Theoretical Distribution
Given the percentiles, we can fit a theoretical distribution. A common choice is the log-normal distribution for positive data or the normal distribution for data that is symmetric.
Fitting a Normal Distribution
The normal distribution can be parameterized by its mean and standard deviation. Using the percentiles, we can estimate these parameters.
# Define the percentiles
p10 <- 5
p50 <- 15
p90 <- 25
# Estimate mean and standard deviation for a normal distribution
mean_est <- p50
sd_est <- (p90 - p10) / (qnorm(0.9) - qnorm(0.1))
cat("Estimated Mean:", mean_est, "\nEstimated SD:", sd_est, "\n")
# Plot the fitted normal distribution
x <- seq(0, 30, length.out = 100)
y <- dnorm(x, mean = mean_est, sd = sd_est)
plot(x, y, type = "l", main = "Fitted Normal Distribution", ylab = "Density",
xlab = "Value")
abline(v = c(p10, p50, p90), col = "red", lty = 2)
Output:
Estimate a Distribution Based on Three Percentiles in R
This plot demonstrates how the normal distribution fits the given percentiles. The estimated mean and standard deviation ensure that the 10th, 50th, and 90th percentiles are at the specified values (5, 15, and 25, respectively). This approach is useful for visualizing and understanding how a normal distribution can be parameterized to match specific percentiles.
Fitting a Log-normal Distribution
If the data is skewed, a log-normal distribution might be more appropriate.
# Transform percentiles for log-normal distribution
log_p10 <- log(p10)
log_p50 <- log(p50)
log_p90 <- log(p90)
# Estimate mean and standard deviation of the log values
meanlog_est <- log_p50
sdlog_est <- (log_p90 - log_p10) / (qnorm(0.9) - qnorm(0.1))
cat("Estimated Meanlog:", meanlog_est, "\nEstimated SDlog:", sdlog_est, "\n")
# Plot the fitted log-normal distribution
x <- seq(0, 30, length.out = 100)
y <- dlnorm(x, meanlog = meanlog_est, sdlog = sdlog_est)
plot(x, y, type = "l", main = "Fitted Log-normal Distribution",
ylab = "Density", xlab = "Value")
abline(v = c(p10, p50, p90), col = "red", lty = 2)
Output:
Estimate a Distribution Based on Three Percentiles in R
This plot demonstrates how the log-normal distribution fits the given percentiles. The estimated meanlog and sdlog ensure that the 10th, 50th, and 90th percentiles are at the specified values (5, 15, and 25, respectively). This approach is useful for visualizing and understanding how a log-normal distribution can be parameterized to match specific percentiles, providing insights into data that is log-normally distributed.
Step 3: Validate the Fit
To validate the fit, we can compare the theoretical percentiles from the fitted distribution to the given percentiles.
# Validate normal distribution fit
qnorm_vals <- qnorm(c(0.1, 0.5, 0.9), mean = mean_est, sd = sd_est)
cat("Theoretical Percentiles (Normal):", qnorm_vals, "\n")
# Validate log-normal distribution fit
qlnorm_vals <- qlnorm(c(0.1, 0.5, 0.9), meanlog = meanlog_est, sdlog = sdlog_est)
cat("Theoretical Percentiles (Log-normal):", qlnorm_vals, "\n")
Output:
Theoretical Percentiles (Normal): 5 15 25
Theoretical Percentiles (Log-normal): 6.708204 15 33.54102
The theoretical percentiles calculated by qnorm using the estimated mean (mean_est) and standard deviation (sd_est) should match the original specified percentiles (5, 15, and 25).
- If they match closely, it confirms that the normal distribution with the estimated parameters accurately represents the specified percentiles.
- The theoretical percentiles calculated by
qlnormusing the estimated meanlog (meanlog_est) and sdlog (sdlog_est) should match the original specified percentiles (5, 15, and 25). - If they match closely, it confirms that the log-normal distribution with the estimated parameters accurately represents the specified percentiles.
The code validates the fitting of normal and log-normal distributions to the specified percentiles. If the theoretical percentiles calculated from the estimated parameters match the original percentiles, it indicates a good fit, confirming that the distributions have been accurately parameterized to match the specified values. This validation step ensures that the fitted distributions are reliable representations of the data described by the given percentiles.
Step 4: Choosing the Best Fit
Compare the theoretical percentiles to the provided percentiles to determine which distribution fits better. The distribution whose theoretical percentiles are closest to the provided percentiles is likely the better fit.
How to Estimate a Distribution Based on Three Percentiles in R?
Estimating a probability distribution based on percentiles is a common task in statistics, especially when dealing with summary data. In this article, we will explore how to estimate a distribution using three given percentiles in the R Programming Language.
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