Statistics (II)
Statistics · Statistics (II)
Syllabus tag: KCSE | Mathematics | Form 4 | Topic 2 Statistics (II)
Lesson objectives
By the end of this topic, you should be able to:
- Calculate the mean using the assumed mean method.
- Make cumulative frequency tables and draw cumulative frequency curves.
- Estimate the median, quartiles and percentiles from the curve.
- Calculate the range, interquartile range, variance and standard deviation.
Statistics (II)
Form 2 found averages. This topic adds efficient methods, position measures, and measures of spread.
a) The assumed mean method
Choose a convenient value near the middle as the assumed mean, A.
For each class find d = x − A, the deviation of the midpoint from A.
Then mean = A + (Σfd ÷ Σf).
The arithmetic stays small because the deviations are small, which is the whole point of the method. The answer is identical to the direct calculation.
Choosing A as the midpoint of the class with the highest frequency usually keeps the numbers smallest.
b) Cumulative frequency
The cumulative frequency is the running total of frequencies.
Each entry is the number of items up to and including that class.
The final cumulative frequency equals the total frequency. If it does not, there is an arithmetic error.
c) The cumulative frequency curve
Plot cumulative frequency against the upper class boundary, not the midpoint. This is the error most often made.
The reason is that the running total is complete only at the top of each class.
Join the points with a smooth curve. The shape is called an ogive.
d) Reading the curve
The median is at the ½n position on the vertical axis. Read across to the curve, then down.
The lower quartile Q₁ is at ¼n, and the upper quartile Q₃ at ¾n.
Percentiles work the same way. The 60th percentile is at 0.6n.
Read the position on the cumulative frequency axis first, then across, then down. Going the other way answers a different question.
e) Measures of spread
The range is the largest value minus the smallest. Simple, but one extreme value distorts it.
The interquartile range is Q₃ − Q₁. It covers the middle half of the data and ignores extremes, which makes it more reliable.
The semi-interquartile range is half of that.
f) Variance and standard deviation
The variance is the mean of the squared deviations from the mean.
Deviations are squared because the plain deviations always sum to zero, so they would measure nothing.
The standard deviation is the square root of the variance. Squaring changed the units, so the root restores them.
A large standard deviation means the data is widely spread. A small one means it clusters near the mean.
The working form Σfx²/Σf − x̄² is usually easier than the definition, since it avoids computing each deviation.
g) Where this is used
Comparing consistency between two sets with the same mean. Quality control, where spread matters more than average. Grading on a distribution.
Words to know
- Assumed mean -- a convenient value chosen to simplify the arithmetic of finding a mean.
- Cumulative frequency -- a running total of frequencies up to a given boundary.
- Ogive -- the cumulative frequency curve.
- Interquartile range -- Q₃ − Q₁, the spread of the middle half of the data.
- Standard deviation -- the square root of the variance, measuring spread in original units.
:::checkpoint Check yourself
- Against what should cumulative frequency be plotted?
- At what position on the curve do you read the median?
- What is the interquartile range?
- Why are deviations squared when finding variance? :::
Bridge to practice
The exercises begin with the assumed mean and cumulative frequency tables, move through the ogive and positional measures, and finish with range, interquartile range and standard deviation. When drawing an ogive, check that your final cumulative frequency equals the total before plotting anything.