Statistics (statistica) - formulario

Descriptive statistics

Definition

• Population: Total collection of all the elements that we are interested in.
• Statistical units: Single elements of the population.
• Variable: Features or characteristics of statistical units.
• Sample: Subgroup of the population that we are able to study in detail.

Variables type

Variables type ↙ ↘

Classes

• Pairs of values that have some relationship to each other → (x, y).
• X qualitative and y quantitative → distinct histograms, one for each category of x.
• X and y quantitative → scatter plot (↗ positive | ↘ negative correlation) →.
• Grouped data: Classes interval use: [ ; ) → “having n1 to n2 means that n1 - n2 is the interval”.
• Take the mid value in order to compute the sample mean [n = mid-value = average of the extremes].

Frequency

Frequency.

Symmetry - ↘↙

X is symmetric: Frequency: Relative frequency:

0- ffrequencies x - c = x + c for any c f = n · w → 0 0 = w.

Mode = m = × → nx.

Sample statistics

Centrality measures and variation - spread

Centrality measures variation - spread - - ↘↓↙ sample median sample mode sample mean sample variance standard deviation.

Sample mean

Sample mean: N ni=1 2 s²Σ x.

(x1 + ... + xn) / n.

× = 1/n Σ xi.

Sample median

Sample median: Order from smallest to larger.

N odd: x → -1 n/2 + 1 n - 1 + 1 2.

N even: x n/2 + x n/2(+1) / 2.

Sample mode

Sample mode: The data value that occurs most frequently.

Sample variance and standard deviation

Sample variance: s² = 1/(n - 1) Σ (xi - ×)².

Standard deviation: s = √s².

Population, covariance and range

Population.

Sample covariance → n n ×y2× → µ maximum - minimum.

x + x Σ x y - k outn Othello n/2 n/2(+1)n ii=1 i+1 Σ f x 2 2i ii=1 s → σ.

ki=1 sxy = 2 2 larger range → l. var. × = = Σ wi xi 2 n-1.

P → p^.

Mean - median relationship and correlation

Mean - median relationship correlation.

sxy Σ xi yi - n · × y.

Symmetric right-skewed left-skewed r = = × = m × > m × < m sx · sy x x x.

Correlation: r = sxy / (sx · sy).

2) y²) x²i (Σ - n · × · (Σ yi² - n ·.

Box-plot

Box-plot - variability index.

1. Median.
2. First-third quartile.
3. IQR.
4. Whiskers.

→ n/2 or n · 0.5.

→ 25 p. = Q1 = n · 0.25 → IQR = 75 p. - 25 p. → LW = 25 p. - 1.5 · IQR.

→ 75 p. = Q3 = n · 0.75 → UW = 75 p. + 1.5 · IQR.

Sample percentiles and linear transformations

Sample percentiles linear transformations - --.

Normal data

Normal data -- Data set normal if histogram has:

• Highest at the middle interval → (mode = sample mean = median).
• Bell-shaped →.
• Symmetry in middle interval →.

Probability theory

Random variables

Random variables.

• Support S: set of possible values which X can take x.
• Discrete random variables: take only integer values.
• Continuous random variables: real (decimal) values.

Probability and density functions

Probability (discrete r.v) function density (continuous r.v) function → →.

Pr(X = x, Y = y) = Pr(X = x) · Pr(Y = y).

f(x,y) = f(x) · f(y) → → X,Y X Y.

Properties

Properties E(·) and Var(·) normal r.v.