Summarising Data: Numerical Methods in Statistics

Summarising Data: Numerical Methods

3.1 Sigma and Factorial Notation

Understanding sigma and factorial notation is fundamental for performing calculations in statistics, especially when dealing with large data sets or combinatorial problems.

3.1.1 Sigma Notation

• Sigma notation (∑) is a concise way to represent the sum of a sequence of terms.

• The general form is $\sum\limits_{i=a}^{b} x_i$, where a is the lower limit, b is the upper limit, and x_i are the terms to be summed.

• Key rules:

  • $\sum\limits_{i=a}^{b} c = (b-a+1)c$

  • $\sum\limits_{i=a}^{b} c y_i = c \sum\limits_{i=a}^{b} y_i$

  • $\sum\limits_{i=a}^{b} (x_i \pm y_i) = \sum\limits_{i=a}^{b} x_i \pm \sum\limits_{i=a}^{b} y_i$

  • $\sum\limits_{i=1}^{n} i = \frac{n(n+1)}{2}$

  • $\sum\limits_{i=0}^{\infty} r^i = \frac{1}{1-r}$, for $|r|<1$

• Example: $\sum\limits_{i=1}^{4} x_i = x_1 + x_2 + x_3 + x_4$

3.1.2 Factorial Notation

• Factorial notation (n!) represents the product of all positive integers up to n.

• Key rules:

  • $n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1$

  • $0! = 1$

• Example: $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$

3.2 Measures of Location

Measures of location describe the central tendency or typical value of a data set. The most common measures are the mean, median, and mode.

3.2.1 Mean

• The mean (average) is the sum of all data values divided by the number of values.

• Sample mean: $\bar{x} = \frac{\sum x_i}{n}$

• Population mean: $\mu = \frac{\sum x_i}{N}$

• Example: For data 46, 54, 42, 46, 32: $\bar{x} = \frac{46+54+42+46+32}{5} = 44$

3.2.2 Median

• The median is the middle value when data are ordered from smallest to largest.

• If n is odd, the median is the middle value; if n is even, it is the average of the two middle values.

• Example (odd n): Data: 32, 42, 46, 46, 54. Median = 46.

• Example (even n): Data: 32, 33, 39, 42, 46, 46, 47, 51, 54. Median = (46+46)/2 = 46.

3.2.3 Mode

• The mode is the value that appears most frequently in the data set.

• Data can be unimodal (one mode), bimodal (two modes), or multimodal (more than two modes).

• Example: Data: 46, 54, 42, 46, 32. Mode = 46.

• For categorical data, only the mode is meaningful.

Identifying the Shape of Data Using Mean, Median, and Mode

• If mean > median > mode: Positively skewed (right-skewed)

• If mean < median < mode: Negatively skewed (left-skewed)

• If mean = median = mode: Symmetric distribution

3.2.4 Percentiles

• A percentile indicates the value below which a given percentage of observations fall.

• To find the pth percentile:

  1. Order the data from smallest to largest.

  2. Compute the index: $i = \frac{p}{100}(n+1)$

  3. If i is not an integer, interpolate between the closest ranks.

• Example: For 12 salaries, the 80th percentile is found at index 10.4, so interpolate between the 10th and 11th values.

3.2.5 Quartiles

• Quartiles divide the data into four equal parts:

• Q1: 25th percentile

• Q2: 50th percentile (median)

• Q3: 75th percentile

• Quartiles are special cases of percentiles and are calculated using the same method.

3.3 Measures of Variability

Measures of variability describe the spread or dispersion of a data set. The simplest measure is the range.

3.3.1 Range

• The range is the difference between the largest and smallest values in the data set.

• Formula: $\text{Range} = x_{\text{max}} - x_{\text{min}}$

• Advantages: Easy to calculate.

• Disadvantages: Only considers two values; highly sensitive to outliers.

• Example: For salaries 3310 to 3925, range = 3925 - 3310 = 615. If one value is 10000, range = 10000 - 3310 = 6690.