Cumulative frequency diagrams — median, quartiles, IQR

Cumulative frequency is plotted at the UPPER class boundary, not the mid-interval value, and the points are joined with a smooth curve — not drawn as bars. Plotting at midpoints shifts the whole curve and throws off every quartile and median read from it.

The interquartile range is $\text{UQ}-\text{LQ}$ (upper quartile minus lower quartile), NOT $\text{UQ}+\text{LQ}$. Adding them gives a number far too large that has no meaning as a spread. The IQR measures the range of the middle 50% of the data, so it must be a subtraction.

The median is read at $\tfrac{n}{2}$ on the curve, the lower quartile at $\tfrac{n}{4}$ and the upper quartile at $\tfrac{3n}{4}$. Reading the lower-quartile value when the median is asked (or mixing up which fraction of $n$ to use) gives the wrong statistic. Match the fraction to the quantity asked for.

For "how many are more than $X$", read the cumulative frequency at $X$ — that is how many are $X$ or LESS — then subtract it from the total. E.g. if $62$ of $80$ are at most $X$, then $80-62=18$ are more than $X$. Give a whole number of items, not a probability.

When finding quartiles from a small ODD-sized list, do not include the median value in both halves. For $11$ values the median is the $6$th; the lower quartile is the middle of the FIRST five ($1$st–$5$th) and the upper quartile the middle of the LAST five ($7$th–$11$th). Leaving the median in both halves shifts both quartiles.

To estimate a median by interpolation, first find the class that contains position $\tfrac{n}{2}$ using the CUMULATIVE totals, then interpolate within THAT class. Choosing a class by frequency alone, or using midpoints, puts you in the wrong interval and gives a wrong median.