Introduction

In this chapter, we explore the concept of estimating quantiles, particularly focusing on the median, using sample data. Quantiles provide crucial insights into the distribution of data and offer a way to understand different percentiles within a dataset. Estimating these quantiles from sample data is an essential skill in statistical analysis, allowing us to make meaningful inferences about a population based on the information at hand.

Quantiles represent specific values in a dataset that divide the data into intervals containing an equal number of observations. The median, often referred to as the second quartile, is a vital quantile that splits the data into two halves, with 50% of the observations lying below and 50% lying above it. Estimating quantiles from a sample provides a way to approximate the corresponding population quantiles and gain insights into the data’s distribution.

Estimating the Median

To estimate the median from a sample, arrange the data in ascending order. If the sample size is odd, the median is the middle value. If the sample size is even, the median is the average of the two middle values. For example, given a sample of exam scores {75, 80, 85, 90, 95}, the median can be estimated as the middle value 85.

Calculating Quartiles

Quartiles divide the data into four quarters, with each quartile representing a specific percentile. The first quartile (Q1) is the 25th percentile, the second quartile (Q2) is the median, and the third quartile (Q3) is the 75th percentile. Estimating quartiles involves finding the values that correspond to these percentiles in the sample.

Interquartile Range (IQR)

The interquartile range is the range between the first quartile (Q1) and the third quartile (Q3). It represents the spread of the middle 50% of the data and is a measure of variability that is not affected by extreme values.

Example: Consider a dataset of ages {20, 25, 28, 30, 32, 35, 40, 42, 45, 50}. To estimate the median, we arrange the data and find the middle value, which is 35. For the quartiles, Q1 is the 25th percentile (25th percentile position is 2.5), so Q1 is 28. Q3 is the 75th percentile (75th percentile position is 7.5), so Q3 is 42. The IQR is then calculated as Q3 - Q1 = 42 - 28 = 14.