How Is the PDF of the Sample Median Derived?

Click For Summary
SUMMARY

The probability density function (PDF) of the sample median can be derived based on whether the sample size is odd or even. For odd sample sizes, the median corresponds to the middle order statistic, allowing for a straightforward derivation of its PDF. For a deeper understanding of asymptotic normality related to the sample median, refer to "Approximation Theorems of Mathematical Statistics" by Serfling, or the texts by Bickel/Dobson and Hogg/Craig for varying levels of rigor.

PREREQUISITES
  • Understanding of probability density functions
  • Familiarity with order statistics
  • Knowledge of asymptotic normality
  • Basic concepts in mathematical statistics
NEXT STEPS
  • Study the derivation of the PDF for the sample median in odd-sized samples
  • Explore "Approximation Theorems of Mathematical Statistics" by Serfling for rigorous statistical approaches
  • Review the texts by Bickel/Dobson and Hogg/Craig for insights on asymptotic normality
  • Investigate the properties of order statistics in probability theory
USEFUL FOR

Statisticians, data analysts, and students of mathematical statistics seeking to understand the derivation of the sample median's probability density function.

natski
Messages
262
Reaction score
2
I am familiar with the derivation of the probability density function of the sample mean but was looking for an equivalent derivation for the sample median. Does anyone know of a good, clear derivation of this PDF?

Thanks,

Natski
 
Physics news on Phys.org
There are two situations. If the sample size is odd, then the median is the middle order statistic and you can immediately write down its pdf.

For the asymptotic normality result, you can look in books on probability or mathematical statistics. Bickel/Dobson have a text that is slightly higher level than Hogg/Craig. For a much more rigorous approach you can look at "Approximation Theorems of Mathematical Statistics" by Serfling.
 

Similar threads

Graduate Recalculate a range by variable 'Median'?
  • · Replies 4 ·
Replies
4
Views
1K
Undergrad What are the commonly used estimators in regression models?
  • · Replies 23 ·
Replies
23
Views
4K
Undergrad How to obtain moment bound from the importance sampling identity?
  • · Replies 1 ·
Replies
1
Views
2K
High School How to get the typical value or typical data from the dataset
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Sampling Weights: Understanding Multipliers & Aggregates
  • · Replies 13 ·
Replies
13
Views
2K
Undergrad Error bars in Excel (standard deviation, standard error, percentage)
  • · Replies 4 ·
Replies
4
Views
3K
Undergrad Desirable estimator properties...
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Proper understanding of p-value
  • · Replies 5 ·
Replies
5
Views
2K
MHB Log likelihood and Maximum likelihood
  • · Replies 1 ·
Replies
1
Views
2K
MHB Sampling distribution of a statistic
  • · Replies 6 ·
Replies
6
Views
3K