Descriptive Statistics: Central Tendency

• Descriptive statistics are numerical measures summarizing data from a sample regarding central tendency and variability.

Measures of Central Tendency

• Central tendency measures summarize the center of a distribution with a single value.
• Common measures: mode, median, and mean.

Mode

• It is the value that occurs most frequently in raw and ungrouped frequency data.
• A variable can have one, two, more than two, or no mode.
• Unimodal signifies one mode.
• Bimodal signifies two modes.
• Multimodal signifies more than two modes.
• For grouped frequency data, identifying the exact mode is impossible, therefore the modal class is used, and the mode is estimated using the midpoint of the modal class.

Median

• The median is the middle value in an ordered dataset, with at most 50% of observations below and 50% above it.
• To find the median for raw data:
• Order the data from lowest to highest.
• Find the median position using the formula: (n+1)/2
• If n is odd, the median position is a whole number. The median value is the value in that position.
• If n is even, the median position is a fraction. The median is the average of the two values on either side of this position.
• Calculating the median for ungrouped frequency tables involves using cumulative frequencies, while grouped frequency tables require cumulative frequencies and interpolation.

Mean

• The mean of a variable is the arithmetic average.
• For raw data, It is found by summing all values and dividing by the number of observations.
• Population mean (μ) formula: μ = (1/N) * Σx
• Sample mean (x-bar) formula: x = (1/n) * Σx
• For ungrouped frequency tables, calculate the mean using a formula based on variable values and their frequencies. Grouped frequency tables estimate the mean using the midpoints of class intervals and their frequencies.

Concluding Notes on Central Tendency Measures

• Mode is valid for categorical and numerical data but can have multiple modes and is not affected by outliers.
• Median is not affected by outliers and is best for skewed data but is only appropriate for ordinal and numerical data.
• Mean is calculated using every value, making it very accurate and best for symmetrical data. However, it is affected by extreme values and only for numerical data.

Measures of Relative Standing

• These measures show where values stand relative to the distribution, using percentiles, which divide data into 100 parts.
• The rth percentile (Pr) separates the lowest r% from the remaining (100 – r)%.
• Interpretation: At most r% is less than Pr, and at most (100 – r)% is more than Pr.
• Other percentiles include deciles (10 equal parts) and quartiles (4 equal parts).
• D5 = Q2 = P50 = median
• To find the percentile value for raw data, sort the data, find the percentile position, and use a formula.

Finding Percentiles for Raw Data

• Order the data from lowest to highest.
• Calculate the percentile position: (r/100) * (n+1).
• The position is in the format k.d, where k is the integer and d is the decimal portion.
• Calculate Pr using the formula: Pr = x(k) + d * (x(k+1) - x(k)), where represents the value in position k of the ordered dataset.

Measures of Variability

• Measures describe how data spreads around central tendency, including range, interquartile range, variance, standard deviation, and coefficient of variation.

Range

• Range is an approximate measure of variability, calculated as maximum - minimum.

Interquartile Range (IQR)

• IQR is the distance between the first and third quartiles (Q3 - Q1).
• It measures the spread of the middle 50% of the data around the median.

Average Deviation

• Average deviation is the arithmetic mean of the differences between each observation and the mean, but the sum always equals zero, making it redundant.

Variance

• It is the average squared deviation around the mean and the most used measure of variability. Larger variance signifies more data variation around the mean.
• Population variance (σ^2) formula: σ^2 = (1/N) * Σ(x - μ)^2
• Sample variance (s^2) formula: s^2 = (1/(n-1)) * Σ(x - x)^2

Standard Deviation

• Standard deviation is the positive square root of the variance.
• Represented in the same units as the variable.
• Population standard deviation (σ) formula: σ = sqrt((1/N) * Σ(x - μ)^2)
• Sample standard deviation (s) formula: s = sqrt((1/(n-1)) * Σ(x - x)^2)

Coefficient of Variation

• Used to compare the variability of different variables, calculated as the ratio of standard deviation to the mean, expressed as a percentage.
• Formula: CV = (s / x) * 100
• This value is not bounded by 100% and can be greater than that value.

Concluding Notes on Variability Measures

• Range is easy to calculate but affected by extreme values.
• Interquartile Range is not affected by outliers but does not use all the data.
• Average Deviation is always zero.
• Variance/Standard Deviation uses available data but is affected by outliers and is best for symmetrical data.
• Coefficient of Variation is the best measure of relative variability and is affected by outliers.

Learn about descriptive statistics, including measures of central tendency such as mode, median, and mean. Explore how to calculate these measures for both raw and grouped data. Understand the concepts of unimodal, bimodal, and multimodal distributions.