Statistics

Abdullah Al Mahmud

Concept

Statistics has three meanings:

  1. Data (table or a series of values)
Light speed experiment
Expt Run Speed
001 1 1 850
002 1 2 740
003 1 3 900
004 1 4 1070
005 1 5 930
006 1 6 850
    1. Plural of statistic (formula)
    1. Method of analyzing and predicting data

Unorganized and Organized Data

40, 39, 31, 38, 40, 40, 34, 39, 31, 38, 37, 30, 31, 37, 35, 37, 36, 35, 39, 39

x Freq
30 1
31 3
34 1
35 2
36 1
37 3
38 2
39 4
40 3

Frequency Distribution

X = 11, 15, 16, 18, 20, 22, 25

Class Tally Frequency
11-15 || 2
16-20 ||| 3
21-25 || 2

Construction

Range = (Highest value - Lowest value) + 1

X = 11, 15, 16, 18, 20, 22, 25

Range of X = ?

Class Interval

\((11-15) \rightarrow (15-11+1)= 5\), not 4

  • \((20-24) \rightarrow ?\)
  • Number of class = \(\frac{Range}{Interval}\)
  • Interval = ?

Example of Frequency Distribution

X = 32, 20, 34, 17, 15, 40, 5, 18, 44, 28, 49, 27, 8, 29, 45, 39, 3, 35, 46, 37, 50, 36, 2, 4, 7, 24, 42, 31, 19, 14

Range = ?

Let, class interval = 5

Number of class = ?

  • Now, construct

Interpretation

Class (Marks out of 40) Frequency
11-15 2
16-20 5
21-25 9
26-30 10
31-35 3
  • What have you known from this frequency distribution?
  • What is the benefit of organizing?

Cumulative Frequency

Class (Marks out of 40) Frequency Cumulative Frequency
11-15 2 2
16-20 5 7
21-25 9 16
26-30 10 26
31-35 3 29
  • Why Useful??
  • How to interpret?

Variable

  • Discrete: Any of the pre-specified number
  • Continuous/Indiscrete: Any number between any two other numbers.
  • Temperature
  • Result of a die throw
  • Mark of a subject
  • GPA of a student
  • Radius of screws

Diagram

Histogram

  • Make sure class intervals are continuous

    Continuous or exclusive: (10-15); (15-20); (20-25) Discontinuous/Inclusive: (10-14); (15-19)

  • If discontinuous \(\rightarrow\) convert
  • Add 0.5 to upper limit and subtract 0.5 from lower limit
\(\downarrow\)

Histogram Example

Interval Frequency
20-30 5
30-40 12
40-50 30
50-60 40
60-70 20
70-80 13
80-90 3
90-100 2

Histogram Example Write its interpretation in 3-5 sentences.

Make a Histogram

Class Interval Continuous CI Frequency
11-20 10
21-30 20
31-40 35
41-50 20
51-60 15
61-70 10
71-80 8
81-90 5
91-100 3

Histogram

Frequency Polygon

  • Locate top midpoints of each bar of histogram
  • Join the points by straight lines.
  • How to draw without drawing a histogram first?
Answer

Frequency Curve

Somoothed version of frequency polygon

Frequency Polygon

Ogives

Cumulative Frequency (CF) for Ogive
intervals CF
0-5 1
5-10 2
10-15 4
15-20 6
20-25 7
25-30 10

Central Tendency

Arithmetic Mean (AM)

  • From Frequency Distribution: Use mid-values and multiply with frequencies

\[AM = \frac 1 n \sum_{i=n}^n f_ix_i \]

AM in Short-cut Method

  • WHY?
  • Is this method really SHORT?

Shortcut Method for AM

Calculate the mean in a smart way

## [1] 1009 1037 1047 1024 1013 1043
Show

Shortcut Method Formula

Consider the values: 1005, 1010, 1015

If 1000 is subtracted: 5, 10, 15

If again divided by 5: 1, 2, 3

Converted Mean = 2

Original Mean = \(2 \times 5 + 1000=1010\)

Show

Short-cut Method on Organized Data