Download presentation

1
**Descriptive statistics (Part I)**

Lecture 2 Descriptive statistics (Part I)

2
**Lecture 2: Descriptive statistics**

Data in raw form are usually not easy to use for decision making Some type of organization is needed Table Graph Techniques reviewed here: Bar charts and pie charts Ordered array Stem-and-leaf display Frequency distributions, histograms Cumulative distributions Contingency tables

3
**Tabulating and Graphing Univariate Categorical Data**

Graphing Data Tabulating Data Pie Charts Summary Table Bar Charts

4
**Summary Table (for an Investor’s Portfolio)**

Investment Category Amount Percentage (in thousands $) Stocks Bonds CD Savings Total Variables are Categorical

5
**Bar Chart (for an Investor’s Portfolio)**

6
**Pie Chart (for an Investor’s Portfolio)**

Amount Invested in K$ Savings 15% Stocks 42% CD 14% Percentages are rounded to the nearest percent Bonds 29%

7
**Organizing Numerical Data**

41, 24, 32, 26, 27, 27, 30, 24, 38, 21 Frequency Distributions & Cumulative Distributions Ordered Array Stem and Leaf Display 3 028 4 1 21, 24, 24, 26, 27, 27, 30, 32, 38, 41 Histograms Tables

8
**The Ordered Array Shows range (min to max)**

Data in raw form (as collected): 24, 26, 24, 21, 27, 27, 30, 41, 32, 38 Data in ordered array from smallest to largest: , 24, 24, 26, 27, 27, 30, 32, 38, 41 Shows range (min to max) May help identify outliers (unusual observations) If the data set is large, the ordered array is less useful

9
**Stem-and-Leaf Display**

A simple way to see distribution details in a data set METHOD: Separate the sorted data series into leading digits (the stem) and the trailing digits (the leaves)

10
Example Data in Raw Form (as Collected): 24, 26, 24, 21, 27, 27, 30, 41, 32, 38 Data in Ordered Array from Smallest to Largest: 21, 24, 24, 26, 27, 27, 30, 32, 38, 41 Stem-and-Leaf Display: 4 1

11
**Tabulating Numerical Data: Frequency Distributions**

What is a Frequency Distribution? A frequency distribution is a list or a table … containing class groupings (ranges within which the data fall) ... and the corresponding frequencies with which data fall within each grouping or category It allows for a quick visual interpretation of the data

12
**Tabulating Numerical Data: Frequency Distributions**

Example: A manufacturer of insulation randomly selects 20 winter days and records the daily high temperature 24, 35, 17, 21, 24, 37, 26, 46, 58, 30, 32, 13, 12, 38, 41, 43, 44, 27, 53, 27

13
Sort Raw Data on days in Ascending Order 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Find Range: = 46 Select Number of Classes: 5 (usually between 5 and 15) Compute Class Interval (Width): 10 (46/5 then round up) Determine Class Boundaries (Limits):10, 20, 30, 40, 50, 60 Count Observations & Assign to Classes

14
**Frequency Distributions, Relative Frequency Distributions and Percentage Distributions**

Data in Ordered Array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Relative Frequency Percentage Class Frequency [10, 20) [20, 30) [30, 40) [40, 50) [50, 60) Total

15
**Graphing Numerical Data: The Histogram**

A graph of the data in a frequency distribution is called a histogram The class boundaries (or class midpoints) are shown on the horizontal axis the vertical axis is either frequency, relative frequency, or percentage Bars of the appropriate heights are used to represent the number of observations within each class

16
**Histogram Example (No gaps between bars) Class Midpoints**

Frequency [10, 20) [20, 30) [30, 40) [40, 50) [50, 60) (No gaps between bars) Class Midpoints

17
**Tabulating Numerical Data: Cumulative Frequency**

Data in Ordered Array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Upper Cumulative Cumulative Limit Frequency % Frequency

18
**Two categorical variables (contingency table)**

The following data represent the responses to a question asked in a survey of 20 college students majoring in business – What is your gender? (Male = M; Female = F) What is your major? (Accountancy = A; Information System = I; Market = M) Gender: M M M F M F F M F M F M M M M F F M F F Major: A I I M A I A A I I A A A M I M A A A I

19
**Contingency table (cont’d)**

Raw data set: Gender: M M M F M F F M F M F M M M M F F M F F Major: A I I M A I A A I I A A A M I M A A A I A I M Total Male 6 4 1 11 Female 3 2 9 10 7 20

20
**Graphical methods are:**

Good in presenting data Not easy for comparison Difficult to use for statistical inference

21
**Numerical description**

Summary Measures Central Tendency (location measures) Quartiles Variation Range Mean Median Mode Variance Interquartile range Standard Deviation

22
**Mean Mean (Arithmetic Mean) of Data Values Sample mean Population mean**

Sample Size Population Size

23
**An example TV watching hours/week: 5, 7, 3, 38, 7**

Mean = ( )/5 = 60/5 = 12 If the correct time for 4th subject is 8 (not 38) Mean = ( )/5 = 30/5 = 6 Mean = 6 Mean = 12

24
Mean (Cont’d) The Most Common Measure of Central Tendency especially when n is large due to its good theoretical properties Affected by Extreme Values (Outliers)

25
**Median Robust measure of central tendency**

Not affected by extreme values In an ordered array, the median is the ‘middle’ number If n is odd, the median is the middle number (i.e,(n+1)/2 th measurement) If n is even, the median is the average of the n/2 th and (n/2 +1) th measurement Median = 7 Median = 7

26
**Mode A Measure of Central Tendency Value that Occurs Most Often**

Not Affected by Extreme Values There May Not Be a Mode There May Be Several Modes Used for Either Numerical or Categorical Data No Mode Mode = 9

27
**Quartiles 25% 25% 25% 25% Split ordered data into 4 quarters**

Position of i-th quartile (1st quartile) and (3rd quartile) are measures of Noncentral Location are called 25th, 50th, and 75th percentile respectively. A pth percentile is the value of X such that p% of the measurements are less than X and (100-p)% are greater than X. 25% 25% 25% 25%

28
**Quartiles (example) Data in Ordered Array: 3 6 6 12 12 12 15 15 18 21**

Position of first quartile is Position of third quartile is

29
**5-number summary Median( ) X X 3 6 12 15.75 21 Box-and-Whisker Plot**

Graphical display of data using 5-numbers Data in Ordered Array: Median( ) X X largest smallest 3 6 12 15.75 21

Similar presentations

© 2022 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google