Statistical Techniques - Study Notes
Lesson summary
Statistics is a branch of science which deals with quantitative data. It includes four processes: collection, presentation, analysis and interpretation of data. Statistical techniques are applied in all fields of academic research; wherever data are collected and summarised or numerical information is analysed, statistics is needed for sound analysis and interpretation of results.
Why geographers use statistics
- To describe and summarise spatial data.
- To make generalisations concerning complex spatial patterns.
- To estimate the probability of outcomes for an event at a given location.
- To use samples of geographic data to infer characteristics of a larger set (population).
- To find whether the magnitude or frequency of a phenomenon differs from one location to another.
- To learn whether an actual spatial pattern matches an expected pattern.
Example: a student studying the annual rainfall of a region first collects the rainfall data, then summarises it in a table or diagram.
Types of data
| Basis | Type | Meaning and example |
|---|---|---|
| Nature | Qualitative | Only describes an item: coconut tree, palm tree, language spoken |
| Nature | Quantitative | Has a numerical value: marks 45, 70, 60, 90, 25 |
| Source | Primary | Collected by the investigator from door to door for his or her study |
| Source | Secondary | Already collected and recorded, taken from published or unpublished sources |
Methods of collection
- Primary: direct personal investigation (interview), indirect oral investigation, questionnaire method.
- Secondary: published sources, unpublished sources.
Organisation and classification of data
Collected data is large in quantity, so it is classified, arranged and reduced before analysis.
- Range = highest value - lowest value. For 2, 6, 4, 9, 12, 8, 5, 8: 12 - 2 = 10.
- Array (frequency array): raw data arranged in ascending or descending order.
- Frequency distribution: arrangement of data into classes.
| Term | Meaning | Class 10 - 20 |
|---|---|---|
| Class | Group dividing the variable into parts with a given frequency | 10 - 20 |
| Class limit | Extreme values of a class | Lower 10, upper 20 |
| Midpoint | (lower limit + upper limit)/2 | 15 |
| Class interval | Upper limit - lower limit | 10 |
| Tally marks | Bars in groups of 5; a cross stroke over four bars for the 5th | - |
| Class frequency | Number of observations in a class | - |
| Total frequency | Aggregate of all class frequencies | - |
The book's tally table has classes 0-10, 10-20, 20-30, 30-40 and 40-50 with frequencies 1, 5, 12, 8 and 4, a total of 30.
Measures of central tendency
An average is a single figure which represents the whole data, for example the heights of 5000 college students. Since it represents the entire data, its value lies between the largest and smallest observations, so it is called a measure of central tendency. The three types of averages are mean, median and mode.
Mean (arithmetic mean)
- Individual values: mean = sum of values / number of items, \( \bar{X} = \Sigma X / N \).
- Example 13.1, Cuddalore monthly temperatures: 341.09 / 12 = 28.42 °C.
- Grouped data: \( \bar{X} = A + \frac{\Sigma fd}{N} \times i \), with A assumed mean, f frequency, i class size, \( d = (x - A)/i \) and N total.
- Example 13.2, plant heights: A = 25, i = 10, d = -1, 0, 1, 2, 3, fd = -5, 0, 10, 30, 15, so \( \Sigma fd \) = 50, N = 50 and mean = 25 + 10 = 35.
Median
- The middle value of a series arranged in ascending or descending order; it divides the series into two equal parts.
- Individual values: median = value of the (N+1)/2 th item.
- Example 13.3, Chennai minimum temperatures: 12 values give the 6.5th item = (24.6 + 25.6)/2 = 25.1.
- Grouped data: find N/2, locate the median class from the cumulative frequency, then median = \( l + \frac{N/2 - m}{f} \times c \).
- l = lower limit of median class, f = its frequency, m = cumulative frequency of the preceding class, c = its class interval.
- Example 13.4, literates by age in 2001: N = 371, N/2 = 185.5, median class 20 - 25, median = 20 + (2.2/73.2) x 5 = 20.15.
Mode
- The value which occurs most frequently, or the item which repeats itself the greatest number of times.
- Grouped data: the modal class is the class with the maximum frequency, and mode = \( l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times c \).
- f1 = frequency of the modal class, f0 = of the preceding class, f2 = of the succeeding class, c = width, l = lower limit.
- Example 13.6, daily wages: modal class 50 - 60, mode = 50 + (3/6) x 10 = 55.
Exam tips
- Always arrange values in order before finding the median.
- For an even number of items, the median is the average of the two middle items.
- The modal class has the highest frequency; the median class is where the cumulative frequency first reaches N/2.