Statistics (I)
Statistics · Statistics (I)
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 18 Statistics (I)
Lesson objectives
By the end of this topic, you should be able to:
- Collect and organise data into a frequency distribution table.
- Group data into reasonable classes.
- Calculate the mean, median and mode for ungrouped and grouped data.
- Represent data using bar graphs, histograms and frequency polygons.
Statistics (I)
Statistics organises data so that a pattern becomes visible, and summarises it in a few numbers.
a) Frequency distribution
A frequency distribution table lists each value or class with how many times it occurs.
Frequency means how many.
Use tally marks in fives while collecting, then convert to figures.
Check that the frequencies sum to the number of items collected.
b) Grouping data
When values are spread widely, group them into classes.
Aim for between five and ten classes. Fewer hides the pattern; more scatters it.
Classes must be of equal width and must not overlap.
The class limits are the stated ends, such as 21 and 30.
The class boundaries are the true dividing lines, 20.5 and 30.5. A mark of 30.4 must belong somewhere.
The midpoint is the average of the limits, so 25.5 for the class 21–30. It represents the whole class in calculations.
c) Mean, median and mode for ungrouped data
The mean is the total divided by the number of values.
The median is the middle value once ordered. With an even count, take the mean of the middle two.
The mode is the most frequent value. There may be more than one, or none.
d) Mean of grouped data
With grouped data the individual values are lost, so the midpoint stands for each class.
Multiply each midpoint by its frequency, sum those products, then divide by the total frequency.
The formula is mean = Σfx ÷ Σf.
The answer is an estimate, because the midpoint is an assumption about where values sit within each class.
e) Median and modal class
For grouped data we name the median class and the modal class rather than exact values.
The modal class has the highest frequency.
The median class contains the middle item, which is the (n/2)th value for grouped data.
f) Choosing an average
The mean uses every value but is dragged by extremes.
The median ignores extremes, so it describes typical values better in skewed data.
The mode is the only average usable for non-numerical data such as favourite colour.
g) Representing data
A bar graph compares separate categories and has gaps between bars.
A histogram shows grouped continuous data. Its bars touch, because the classes are continuous. Bar width represents the class width.
A frequency polygon joins the midpoints of the tops of the histogram bars with straight lines. It is useful for comparing two distributions on one diagram.
Always start the frequency axis at zero and label both axes.
h) Where this is used
Exam analysis. Census reporting. Quality control. Any decision that rests on a body of collected data.
Words to know
- Frequency -- the number of times a value or class occurs.
- Class boundaries -- the values midway between consecutive classes, used for histograms.
- Midpoint -- the average of a class's limits, representing that class.
- Cumulative frequency -- a running total of frequencies.
- Modal class -- the class with the highest frequency.
:::checkpoint Check yourself
- What is the midpoint of the class 31–40?
- Why does a histogram have no gaps between its bars?
- Find the mean of 4, 7, 7, 10, 12.
- Why is the mean of grouped data only an estimate? :::
Bridge to practice
The exercises begin with frequency tables and grouping, move through the three measures for ungrouped and grouped data, and finish with representation and choosing an appropriate measure. When finding a median, write the ordered list out in full before locating the middle.