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Statistics has three meanings:

- Data (table or a series of values)

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 |

- Plural of statistic (formula)

- Method of analyzing and predicting 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 |

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

Class | Tally | Frequency |
---|---|---|

11-15 | || | 2 |

16-20 | ||| | 3 |

21-25 | || | 2 |

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 = ?

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

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?

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?

- 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

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

\(\downarrow\)

Conversion

Continuous CI | Discontinuous CI |
---|---|

10-14 | 9.5-14.5 |

15-19 | 14.5-19.5 |

20-24 | 19.5-24.5 |

25-29 | 24.5-29.5 |

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 |

Write its interpretation in 3-5 sentences.

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

- Locate top midpoints of each bar of histogram
- Join the points by straight lines.

- How to draw without drawing a histogram first?

Answer

- Find midpoints of class intervals
- Join the tops of all frequencies.

Somoothed version of frequency polygon

intervals | CF |
---|---|

0-5 | 1 |

5-10 | 2 |

10-15 | 4 |

15-20 | 6 |

20-25 | 7 |

25-30 | 10 |

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

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

- WHY?
- Is this method really
*SHORT*?

Calculate the mean in a smart way

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

Show

Subtract a number from all, say 1020

`## [1] "The new values are"`

`## [1] -11 17 27 4 -7 23`

`## [1] "Mean of y is 8.83"`

`## [1] "Mean of x is 1028.83"`

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

x = 1005, 1010, 1015

- a = 1000
- c = 5
- y = 1, 2, 3
- \(\bar x = 2 \times 5 + 1000=1010 = a+\bar y \times c\)
- \(\bar x = a+\frac{\sum y}{n} \times c\)