Descriptive Statistic

Large N & Leeuwenhoek (70700173)

Yue Hu

Overview

• Data

• Variable
  • Typology
  • Point
  • Dispersion

• Description
  • Point
  • Dispersion

1 Data

1.1 Concept

Common definition:

The quantities, characters, or symbols on which operations are performed by a computer, being stored and transmitted in the form of electrical signals and recorded on magnetic, optical, or mechanical recording media.

Oxford Language:

Things known or assumed as facts, making the basis of reasoning or calculation.

Etymology

/ˌɛtɪ’mɑlədʒi/

from “a piece of info, something to give” to now

1.2 Think deeper

• Is data a product?
• Is data the “new oil”?
• Is noise data?

• Data is a type of project, but a special one among private, club, common resources (exclusiveness and rivalry), between the club and public goods
• Difference from oil
  • Unlimited
  • Can’t valued
  • Replicable
• Noise is not data
  • Data’s value is from analysis

If you can’t measure it, you can’t manage it

If you don’t understand it,
you can’t well measure it.

2 Variable

2.1 Definition

Specifically, a variable = a dimension(well…) = a column

Empirically, a variable = a piece of data with homogeneous properties

Relationally, an observation ⊂ {var1, var2, var3,…} ⊂ a dataset

2.2 Quiz

A concept?
A variable?
A measurement?

2.3 Typology

• Constant vs. variables
• Dependent/outcome/response ~
• Independent/explanatory/treatment ~

When would you want a variable to be constant in a research?

1. Get rid of the effect
2. Testing if X is a necessary condition for Y
   • Is hard work necessary for successes?
   • A works 12h, B works 12h, A success, B doesn’t
   • Is good governance produced by democracy?
     • Democracy with Trump
     • Democracy with Clinton

Nominal/categorical (discrete) < Ordinal (discrete) < Numeric(continuous)

• Interval: Meaningful distance
  • Discrete/continuous
  • Zero is a terminal
  • Meaningful addition/subtraction

• Ratio: Interval with meaningful zero
  • Continuous
  • Zero means absence
  • Meaningful multiplication/division

feeling thermometer: 0-200，冷淡到热情, 0是非常极端的情绪，不代表没有, so an interval; -100-0-100， 0代表0的意思，ratio 是唯一一种scale可以表示meaningful ratio

Ratio: the ratio of two points on the scale conveys the same meaning regardless of where one is on the scale

e.g.1, Public budget, safty:health = 4:1 then no matter how much in total, it will 4:1 e.g.2, event history, vaccine rate

2.4 Interval vs. Ratio

2.5 Summary

Data

• Fact(-ish)…
• Product…
• Foundation…

Variable

• Definition: a dim…
• Terminology:
  • Y: …
  • X: …
• Typology
  • Discrete…
  • Continuous…

Data

• Fact(-ish) for reasoning
• Product with value
• Foundation of effective governance

Variable

• Definition: a dimension
• Terminology:
  • Y: Outcome/reponse
  • X: Explanatory/treatment
• Typology
  • Nominal/categorical < Ordinal < Numeric (interval/ratio)

3 Description

3.1 Level of description

Raw data

Time series plots, scatter plots

Histogram

Frequency distribution (_i, n_. = ∑n_i, n_.. = ∑n_ij)

Point estimations/Moments

Information decay

3.2 Moment: Point

• Mode: most frequent occurring number, not good of central tendency, bimodal (two modes)
• Median: at 50%
  • Odd: middle value
  • Even: mean of two mid values
  • Sensitive for outliers
• Mean: \(\bar{X}\rightarrow\mu.\)
  • Group: \(\bar{X} = 1/n\sum xf.\), x is the cell midpoint and f is the frequency of \(X_i\)

Note

• For symmetric distributions, median and mean are identical
• Point → “common,” “usual,”“average”

3.3 Think deeper

I rated the applicant among the top 10-15% students taking BSc-courses at Manchester; and now I know for sure that she is among the best…

What’s the relationship between percentile and mean?

A prospective student graduated from Manchester and Cambridge

Information lost increasingly more

percentile is mean of a binary variable

The percentage is the 95% above is 1 and 95% off is 0 and get the mean

3.4 When to use what

Hint

What’s the “point” estimate for?

3.5 How to measure it

3.6 Moment: Dispersion

Range: Max - min; [min, max]

Q: What’s the range of 2, 2, 2, 2?

• Interquartile Range (IQR): [25% quartile, 75% quartile]
• Mean absolute deviation (MAD): \(1/n\sum|X - \bar{X}|.\)
• Mean square deviation (MSD): \(1/n\sum(X - \bar{X})^2.\)

MAD: Intractable, difficult in differentiate absolute values

Dispersion means how wild one can be

↪ Variance: σ² for the population;¹ s² for a sample²

\[\begin{align} \sigma^2 =& \sum(X - \mu)^2p(X);\\ s^2 =& \sum(X - \bar X)^2p(x), \\ =& \frac{\sum(X - \bar{X})^2}{n - 1}. \end{align}\]

σ: sigma

finite: finite

• Population
• Sample

Why n - 1?

1. The entire group you care (finite).

2. The group you collect data

3.7 Why n - 1

Intuitively, Only when there are more than one items, the variance can exist.

In statistics, degree of freedom (d.f.)^*1

\[\begin{equation} \begin{bmatrix} X_1\\ \vdots \\ X_n \end{bmatrix} = \bar{X} \begin{bmatrix} 1 \\\vdots \\ 1 \end{bmatrix} + \begin{bmatrix} X_1 - \bar{X} \\ \vdots \\ X_n - \bar{X} \end{bmatrix} \end{equation}\]

The first vector on the right-hand side is constrained to be a multiple of the vector of 1’s, and the only free quantity is \({\bar {X}}\). It therefore has 1 degree of freedom.

The second vector is constrained by the relation \({\displaystyle \sum _{i=1}^{n}(X_{i}-{\bar {X}})=0}\). The first n − 1 components of this vector can be anything. However, once you know the first n − 1 components, the constraint tells you the value of the nth component. Therefore, this vector has n − 1 degrees of freedom.

In statistical testing applications, often one isn’t directly interested in the component vectors, but rather in their squared lengths.

1. For a crosstable: (r - 1)(c - 1)

3.8 Two Formats of Variance

Definition：

\[\begin{align} \sigma^2 =& \sum(X - \mu)^2p(X);\\ s^2 =& \sum(X - \bar X)^2p(x), \\ =& \frac{\sum(X - \bar{X})^2}{n - 1}. \end{align}\]

Given μ = ∑Xp(x), ∑p(x) = 1,

\[\begin{aligned} \sigma^2 =& \sum(X - \mu)^2p(x),\\ =& \sum(X^2 -2X\mu + \mu^2)p(x),\\ =& \sum X^2p(x) - 2\mu\sum Xp(x) + \mu^2\sum p(x). \\ =& \sum X^2p(x) - \mu^2. \end{aligned}\]

NB: Pay attention to the .red[punctuation marks].]

3.9 Standard deviation (SD)

• Population: \(\sigma = \sqrt{\sum(X - \mu)^2p(X)}\);
• Sample: \(s = \sqrt{\frac{\sum(X - \bar{X})^2}{n - 1}}\);
• Ratio: \(\sigma = \sqrt{\pi(1 - \pi)}\);
• Group: \(\sqrt{\frac{f}{n}\sum(X - \bar{X})^2}\).

3.10 Take-home point

4 R Lab

4.1 Learning R with Dr. Hu

Starting from math💀

4.2 Hey, no fear

Launch (the Rstudio)!