Statistics is the science of collecting, organising, presenting, analysing and interpreting numerical data to aid decision-making.
- Key Terms:
- Primary data: Data collected directly by the investigator (e.g. surveys, experiments).
- Secondary data: Data collected from existing sources (books, websites, reports).
- Raw data: Data in its original, unorganised form.
- Frequency: The number of times a value (or class) occurs.
- Class interval: A range of values grouped together; class width = upper limit - lower limit.
- Class mark (midpoint): (Upper limit + Lower limit) / 2.
- Measures of Central Tendency:
- Mean (arithmetic mean): Sum of all observations / number of observations. x-bar = (sum of xi) / n.
- Median: The middle value when data is arranged in order. For n observations: if n is odd, median = ((n+1)/2)th term; if n is even, median = average of (n/2)th and ((n/2)+1)th terms.
- Mode: The observation that occurs most frequently.
- Graphical Representation:
- Bar graph: Bars of equal width, lengths proportional to frequency; gaps between bars for discrete data.
- Histogram: Similar to bar graph but for continuous data; no gaps between bars.
- Frequency polygon: Midpoints of tops of histogram bars joined by line segments.
Find the mean of 4, 7, 11, 13, 15.
Mean = (4+7+11+13+15)/5 = 50/5 = 10.
The marks of 9 students are: 12, 16, 20, 20, 22, 24, 28, 30, 32. Find the median.
n = 9 (odd). Median = ((9+1)/2)th = 5th term = 22.
Find the mode of: 3, 5, 7, 5, 3, 5, 8, 5, 3.
5 appears 4 times (most frequent). Mode = 5.
The mean of 5 numbers is 18. If one number is removed and the new mean is 20, what was the removed number?
Original sum = 5 x 18 = 90. New sum = 4 x 20 = 80. Removed number = 90 - 80 = 10.
Data: 15, 25, 35, 45, 55. Find mean using assumed mean method with A = 35.
Deviations (di): -20, -10, 0, 10, 20. Sum of di = 0. Mean = A + (sum of di)/n = 35 + 0/5 = 35.
Construct a frequency distribution table for the data: 10, 12, 11, 10, 14, 12, 13, 11, 12, 10.
10: 3 times; 11: 2 times; 12: 3 times; 13: 1 time; 14: 1 time. Total frequency = 10.
Find the class mark of the class interval 20-30.
Class mark = (20 + 30)/2 = 25.
Common mistakes
Mean is affected by extreme values (outliers); median and mode are more resistant. Students sometimes forget to arrange data in order before finding the median. Also, the class mark (midpoint), not the class boundaries, is used as representative value in calculations.
Summary
Statistics enables data analysis through measures of central tendency (mean, median, mode) and graphical tools (bar graphs, histograms, frequency polygons). Each measure has specific advantages depending on the data type and research question.