AI Tutor
If a government census records the salaries of 100,000 citizens in $10,000 bins, you lose every individual's exact paycheck. How can you accurately pinpoint the true center of the population from grouped summaries alone?

Raw Data: Exact Centers

For raw observations, the sample mean (xˉ\bar{x}) represents the physical balance point of the dataset, calculated by dividing the sum of all values by the sample size nn.
The median is the physical middle value of an ordered dataset, located at position n+12\frac{n+1}{2}.
Raw data provides exact precision, but large-scale experiments and survey reports frequently present data organized into intervals. How do we calculate the center when individual values are hidden?

Grouped Data: Estimating the Mean

Because raw values inside a class interval are unknown, we make the simplifying assumption that every observation in a class sits exactly at the class midpoint (mi=lower+upper2m_i = \frac{\text{lower} + \text{upper}}{2}).
The estimated mean weights each midpoint mim_i by its corresponding class frequency fif_i: xΛ‰β‰ˆβˆ‘fimiβˆ‘fi\bar{x} \approx \frac{\sum f_i m_i}{\sum f_i}
While the mean clusters all values at interval midpoints, how can we slice into an interval to locate the 50th percentile?

Grouped Data: Linear Interpolation for the Median

For continuous grouped data, we find the median class containing the n2\frac{n}{2}-th item and assume the frequencies spread evenly across that entire interval.