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Descriptive Statistics: Mean, Median, Mode & Standard Deviation

108 cards · 🇺🇸 English

Measures of central tendency and spread for describing a dataset.

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Descriptive statistics

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1Descriptive statistics
The process of summarizing and describing features from a collection of data, rather than making inferences about a population
2What distinguishes descriptive from inferential statistics?
Descriptive statistics summarize a sample; inferential statistics use data to learn about the population it represents
3Measures of central tendency
Statistics that describe the "center" or typical value of a dataset (mean, median, mode)
4Measures of dispersion (variability)
Statistics that describe the spread of data values (range, variance, standard deviation, IQR)
5Arithmetic mean
The sum of all values divided by the number of values: x̄ = (Σx)/n
6Median
The middle value when data are arranged in order; 50% of values are below and 50% above
7Mode
The most frequently occurring value in a dataset
8Geometric mean
The nth root of the product of n values: GM = (x₁ · x₂ · ... · xₙ)^(1/n), useful for growth rates
9Harmonic mean
HM = n / (Σ(1/xᵢ)), useful for averaging rates like speed
10Relationship AM ≥ GM ≥ HM
For nonnegative real numbers, arithmetic mean ≥ geometric mean ≥ harmonic mean (equality holds when all values are equal)
11Range
The difference between the maximum and minimum values: Range = max − min
12Variance
The average of the squared deviations from the mean: σ² = Σ(x − μ)² / N (population) or s² = Σ(x − x̄)² / (n−1) (sample)
13Standard deviation
The square root of variance: σ = √(variance), measuring spread in the same units as the data
14Population standard deviation formula
σ = √[Σ(x − μ)² / N], where μ is the population mean and N is population size
15Sample standard deviation formula
s = √[Σ(x − x̄)² / (n−1)], using Bessel's correction (n−1) for unbiased estimation
16Why use (n−1) instead of n in sample standard deviation?
Bessel's correction provides an unbiased estimate of the population variance from a sample
1768-95-99.7 rule (empirical rule)
For normal distributions: ~68% within 1σ, ~95% within 2σ, ~99.7% within 3σ of the mean
18What does a high standard deviation indicate?
Data points are spread far from the mean; high variability
19What does a low standard deviation indicate?
Data points are clustered closely around the mean; low variability
20Quartiles
Values that divide data into four equal parts: Q1 (25th percentile), Q2 (median), Q3 (75th percentile)
21Interquartile range (IQR)
IQR = Q3 − Q1, the range of the middle 50% of data, robust to outliers
22Percentile
A value below which a given percentage of observations fall (e.g., 90th percentile = 90% below)
23Skewness
A measure of the asymmetry of a distribution; positive (right skew) or negative (left skew)
24Kurtosis
A measure of the "tailedness" or sharpness of a distribution's peak
25Univariate analysis
Describing the distribution of a single variable (central tendency, dispersion, shape)
26Bivariate analysis
Describing the relationship between two variables (correlation, scatterplots, cross-tabulations)
27Histogram
A graphical representation of the distribution of numerical data using bins/bars
28Box plot (box-and-whisker plot)
A graphical display showing median, quartiles, and outliers of a dataset
29Stem-and-leaf display
A visual representation showing the distribution while preserving individual data values
30Scatterplot
A graph showing the relationship between two variables as points in 2D space