Overview
This lecture covers the basics to advanced concepts in Statistics, focusing on measures of central tendency (mean, median, mode), types of data, and measures of dispersion (range, mean deviation, standard deviation, variance) with calculation methods and interpretation.
Central Tendency: Mean, Median, Mode
- Central tendency is represented by single values summarizing data: mean (average), median (mid-value), and mode (most frequent value).
- Mean is the sum of all observations divided by the number of observations.
- Median is the middle value when data is ordered; for odd n: (n+1)/2th value, for even n: average of (n/2)th and (n/2)+1th values.
- Mode is the value with the highest frequency in data.
Types of Data & Frequency Distributions
- Data types: Raw data (ungrouped), discrete frequency distribution, and continuous (grouped) frequency distribution.
- In discrete frequency, observations may repeat and are listed with their frequencies.
- Continuous frequency uses class intervals and frequencies, requiring calculation of class marks (mid-values).
Measures of Dispersion
- Dispersion measures how spread out data is from the central value.
- Range = Highest observation – Lowest observation; not a good measure as it only considers extremes.
- Mean deviation is the average absolute difference between each observation and the mean/median.
- Calculate mean deviation by summing all absolute deviations and dividing by the total number of observations.
- Standard deviation measures average distance from the mean without using absolute values but by squaring differences.
- Variance is the mean of squared deviations from the mean; standard deviation is the positive square root of variance.
Calculation Methods
- For raw data: compute mean, deviations, squares, sum and average as per formulas.
- For discrete/continuous data: use frequency-weighted formulas, and shortcut methods (step deviation/UI method) to simplify calculations.
- When adjusting all data points (adding or multiplying by a constant), mean and variance change predictably: add/multiply constants to mean; variance is unaffected by addition but multiplied by square of constant if multiplied.
Key Terms & Definitions
- Observation — Each individual data point.
- Mean (x̄) — Sum of observations divided by total number.
- Median — Middle value in ordered data set.
- Mode — Most frequent value in the data set.
- Range — Difference between the largest and smallest value.
- Deviation — Difference of each observation from mean or median.
- Mean Deviation — Average of absolute deviations from mean/median.
- Variance (σ²) — Mean of squared deviations from the mean.
- Standard Deviation (σ) — Positive square root of variance.
- Class Mark — Midpoint of a class interval: (Lower limit + Upper limit)/2.
- Frequency (f) — Number of times an observation/class occurs.
Action Items / Next Steps
- Practice homework problems converting (inclusive/exclusive) class intervals as discussed.
- Solve example questions for mean, median, mode, range, mean deviation, and standard deviation for all data types.
- Review and memorize key formulas for central tendency and dispersion.