05 - Descriptive Statistics

Descriptive Statistics

frequency distributions

• records number of times each value occur

arithmetic mean

• sum of values of items / number of items

the mode

• Definition: value that appears most often in a set of data
• In a grouped frequency distribution
  • define the modal class (class with largest frequency)
  • use the formula: \begin{equation} \text{Mode} = L + \frac{{d_1}}{{d_1 + d_2}} \cdot h \end{equation} $L$: the lower boundary of the modal class $d1(d2)$: absolute value of the difference between the frequency of the class modal and the class before (after) it $h$: length of the interval of the modal class.

the median

• Definition: the number in the middle of a sorted data set.
  • even-size data set: average of 2 middle numbers
  • odd-size data set: middle number

Range:

• the difference between the largest and the smallest value

Percentile:

• a measure indicates value BELOW which a given percentage of observations in a group of observations falls
  • eg. 20th percentile = value below which 20% of observations found
• Find location of the Pth percentile $Lp=(n+1)\cdot \frac{P}{100}$ $L$: location in the ordered array of the desired percentile $n$: number of observations $P$: desired percentile
• Find percentile based on location:$Pth=valueatlowerLp+P\cdot value(higher-lower)Lp$

Quartiles

• Definition: three points divide dataset into 4 equal groups, each comprises a quarter of the data
• $Q_1=P_{25},Q_2=P_{50},Q_3=P_{75}$ $Q_1$ : middle number btw smallest & median $Q2$: median $Q3$: middle value btw median & highest

Deciles

• any of nine values divide sorted data into 10 equal parts, each represents 10% of the sample / population
• just like Percentile, for eg 8th Decile = 80% Percentile

Percentile with grouped data

• calculate from frequency table $loc$: from $P_n$ calculate the location → determine the class from the cumulative frequency (round up) $P_n=firstclassvalue+(lastcumfreqtothisloc)\cdot \frac{classrange}{frequency}$

Inter-quartile range (IQR)

• $IQR=Q_3-Q_1=P_{75}-P_{25}$

Variance

• the average of the squared mean deviation for each value in a distribution
• normal variance
  $x_i$: individual observation $\bar{x}$ : population mean $n$: number of observations
• Variance with grouped data
  $f$: class frequency $M$: class midpoint $n$: number of observations

Standard deviation:

• square root of the variance $$\sigma = \sqrt{\sigma^2}

- Standard deviation: in finance, can measure the risk associated with various investment opportunities. the higher the std, the greater the risk - Two investments with similar std may have different distribution of returns <!--ID: 1708098043547--> ## symmetry - one that can be divided into two mirrored halves of each other, with same arithmetic mean, median, and mode ("bell curve") <!--ID: 1708098043549--> ## skewness - the asymmetry of a frequency distribution curve, with different mean, mode, median - positive skewed distribution: lean left (just like holding left thumb), $mean > median$ - negative skewed distribution: lean right (just like holding right thumb), $mean < median$ <!--ID: 1708098043553--> ## hard parts -> cheat sheet - interpretation of the Quartiles <!--ID: 1708098043557-->