7.1 Dispersion and Frequency Tables for Grouped Data
Class Intervals and Related Terms
Grouped data is data organized into continuous ranges called class intervals. Key definitions include:
- Lower Limit: The smallest value in a class interval.
- Upper Limit: The largest value in a class interval.
- Lower Boundary: $\frac{\text{Upper limit of previous class} + \text{Lower limit of current class}}{2}$
- Upper Boundary: $\frac{\text{Upper limit of current class} + \text{Lower limit of next class}}{2}$
- Class Midpoint ($x$): $\frac{\text{Lower limit} + \text{Upper limit}}{2}$
- Class Size (Width): Upper boundary - Lower boundary
Histograms and Frequency Polygons
- Histogram (with equal class size): Bar chart where the x-axis represents boundaries or midpoints and the y-axis represents frequency. Bars are contiguous (no gaps).
- Frequency Polygon: Line graph formed by connecting midpoints of the tops of histogram bars. Includes two additional midpoints on the x-axis with frequency zero at both ends.
7.2 Cumulative Frequency and Ogives
Cumulative Frequency and Ogive Construction
Cumulative frequency is the running total of frequencies up to the upper boundary of a class interval.
- An Ogive (cumulative frequency curve) is drawn by plotting cumulative frequency against upper boundaries.
- The curve starts at the lower boundary of the first class interval with a cumulative frequency of $0$.
Percentiles and Measures of Position from an Ogive
From an ogive representing total frequency $N$:
- First Quartile ($Q_1$): Value at $\frac{1}{4}N$ (25th percentile, $P_{25}$)
- Median ($Q_2$): Value at $\frac{1}{2}N$ (50th percentile, $P_{50}$)
- Third Quartile ($Q_3$): Value at $\frac{3}{4}N$ (75th percentile, $P_{75}$)
- Interquartile Range ($IQR$): $Q_3 - Q_1$
- $k$-th Percentile ($P_k$): Value at $\frac{k}{100}N$
7.3 Measures of Dispersion (Variance and Standard Deviation)
Mean of Grouped Data
For grouped data with midpoints $x$ and corresponding frequencies $f$:
$$\bar{x} = \frac{\sum fx}{\sum f}$$
Variance ($\sigma^2$) and Standard Deviation ($\sigma$)
The variance and standard deviation measure how spread out data points are from the mean:
$$\text{Variance } (\sigma^2) = \frac{\sum f(x - \bar{x})^2}{\sum f} = \frac{\sum fx^2}{\sum f} - \bar{x}^2$$
$$\text{Standard Deviation } (\sigma) = \sqrt{\frac{\sum fx^2}{\sum f} - \bar{x}^2}$$
Comparing Distributions
- A smaller standard deviation indicates that the data is concentrated closely around the mean (more consistent).
- A larger standard deviation indicates greater dispersion or variation in the dataset.