Statistics

Summation

Descriptive statistics with code in Python

Descriptive statistics

• Divide into
  • Measures of central tendency
    • Answers “What does the middle of our data look like?”
      • Mean
      • Median
      • Mode
  • Measures of spread
    • Answers “How much does my data vary?”
      • Range and interquartile range
      • Standard deviation
      • Variance

Mean

• Average value of a data set
• Sum of all values divided by the number of observations

Median

• Define a typical value in the data set
• Does not require calculation
• Value that coincides with the middle of the data set

Mode

• Value that appears the most frequently in our data
• Intuition of the mode as the “middle” is not as immediate as mean or median, but there is a clear rationale
• Highest weighted contributing factor to our mean

Range and interquartile range

• Range
  • Maximum - minimum value
• Interquartile range (IQR)
  • Also called the midspread or middle 50%, or technically H-spread, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles, IQR = Q3 − Q1

Standard deviation

• Measure of the spread of your observations
• Statement of “how much your data deviates from a typical data point”
• Summarizes how much your data differs from the mean

Variance

• Square of the standard deviation, for the reason of:
  • Avoiding negative values in the sum
  • Pointing out the significance of outliers
  • Having an exponential term that allows us to find where the point of minimum deviation is
• Usually it is enough to give mean and standard deviation, but it is good to note variance as well

Sigmoid function

• Classify as 1/0 (yes/no)
• Give probability, for example, “there is 65% probability of ‘yes’”
• Video

Derivative

• Derivative at a point ← slope of the straight line tangent to f(x) at a chosen value x
  • More definition
• Derivative of f(x) is
  • Slope of f(x)
  • Instantaneous rate of change of f(x)
• Calculating derivative
  • $f(x) = x^{10}$
  • $f'(x) = 10x^9$ ← typical derivative
  • More examples
• Uses of derivatives
  • Find minima
  • Find maxima
  • Find inflection points

Integral

• Matemaks (całka nieoznaczona)

• S (darker colored section) ← integral of f(x) in the interval from “a” to “b”
• $\int_{a}^{b}f(x)dx$
• To calculate: we need to find equation, from which derivative would equal f(x)
• Important formulas
  • $\int x^n dx=\frac{1}{n+1}x^{n+1}+C$ ← for $n \neq -1$
    • because $\int x^{-1} \, dx=\frac{1}{0}x^0$ ← that is not possible
  • $\int \frac{1}{x} \, dx=\ln|x| + C$ ← can be used for $n=1$
• Examples:
  • $\int x^2 \, dx=\frac{1}{3}x^3 + C$
    • because $\left( \frac{1}{3}x^3 \right)'=\frac{1}{3}\times 3x^2=x^2$
    • C = any constant
  • $\int 5x^7 \, dx=5\int x^7 \, dx=5\times \frac{1}{8}x^8+C$
  • $\int \frac{x^2+x^7}{x^3} \, dx=\int \left( \frac{1}{x}+x^4 \right) \, dx=\int \frac{1}{x} \, dx+\int x^4 \, dx=\ln|x|+\frac{1}{5}x^5+C$ ← it’s enough to type one C
  • $\int \left( 5+7x^{\frac{-1}{4}} \right) \, dx=\int 5 \, dx+7\int x^{\frac{-1}{4}} \, dx=5x+7 \times \frac{4}{3}x^{\frac{3}{4}}+C$
    • imagine that $5=5\times x^0$ ← but it’s not normal to think like that
  • $\int \cos(x) \, dx=\sin (x)+C$
  • $\int \sin(x) \, dx=-\cos(x)+C$