Mean of grouped data
For grouped data we cannot use the raw values, so we use the class midpoint $x$ to represent each class. Multiplying each midpoint by its frequency $f$ and summing gives the estimated mean:
Key formula
$\bar{x}=\frac{\sum fx}{\sum f}$
Variance and standard deviation
The variance measures how spread out the data are about the mean. The standard deviation is its square root and has the same units as the data.
Key formula
$\text{variance}=\frac{\sum fx^2}{\sum f}-\bar{x}^2,\qquad \text{standard deviation}=\sqrt{\text{variance}}$
Example
Suppose $\sum f=10$, $\sum fx=200$ and $\sum fx^2=5000$. Then $\bar{x}=\dfrac{200}{10}=20$, variance $=\dfrac{5000}{10}-20^2=500-400=100$, and standard deviation $=\sqrt{100}=10$.
Interquartile range from an ogive
From the ogive we read the first quartile $Q_1$ at cumulative frequency $\tfrac{n}{4}$ and the third quartile $Q_3$ at $\tfrac{3n}{4}$. The interquartile range is $Q_3-Q_1$; a larger value means the data are more spread out.
Remember
Standard deviation and interquartile range both measure dispersion; a smaller value means the data are more consistent.