Probability and Statistics for Engineers: Stem-and-Leaf Plots and Histograms - Prof. Yicha, Study notes of Statistics

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STAT 370: Probability and Statistics for

Engineers^ Yichao Wu

North Carolina State University

Lass class

-^ Factor, level, (complete) factorial study, fractionalfactorial study, valid, accurate, precise, •^ Simple random sampling •^ Stem-and-leaf plot Homework #1 due Friday 16 at 5 PM

• Example: Midterm Scores of STAT
  • The following data setcontains the midterm exam scores of STAT 101.
    • Its stem-and-leaf display^3 645 036 00227 0123446778899998 023334445556677789 35555710

Back-to-back Stemplot

Number of home runhits each season • Babe Ruth (New YorkYankees): 1920-1934 –^ 54 59 35 41 46 25 4760 54 46 49 46 41 34^22 • Mark McGwire (St.Louis Cardinals):1986:2001 –^ 49 32 33 39 22 42 9 939 52 58 70 65 32 29

Its back-to-back stemplot

9766611 4 29^ 944 5

Spending (in dollars) at a supermarket

Example: A study on litter size (mice)

-^ Litter size: the number of offsprings produced atone birth by an animal. •^ Data: (170 observations)^ 4 6 5 6 7 3 6 4 4 6 4 4 9 5 10 6 6 5 68 2 7 7 7 9 3 7 5 7 7 4 5 5 6 7 6 7 86 6 7 6 6 7 5 4 5 6 6 1 3 4 7 5 4 7 58 8 5 6 8 5 5 4 9 6 7 3 7 7 5 4 6 9 67 7 5 7 3 7 6 5 3 7 10 5 6 8 7 5 5 7 55 8 9 7 5 7 5 5 5 6 3 7 8 7 7 6 3 4 44 7 2 7 8 5 8 6 6 5 6 4 7 5 5 6 9 3 54 8 3 9 8 3 6 5 4 7 8 4 8 6 8 5 6 4 38 8 6 9 5 5 6 6 7 6 8 6 11 6 5 6 6 3

Limitations of Stemplot

-^ Awkward for large data sets •^ Splitting stem/rounding is not very helpful.

Histogram (how?)

• Choose intervals (or bins) that cover the entire range ofthe data –^ Count the number of observations per interval – Draw rectangles with heights corresponding to number ininterval (for relative frequency histogram, height is relativefrequency of interval) Notes: •^ Intervals must be non-overlapping •^ Typically, an observation equal to a boundary value is put in thehigher interval. •^ Intervals must be contiguous (rectangles touch each other) •^ Intervals must be equal width •^ Often choose “nice” boundaries

Example

Example: The manager at Wendy’s is interested in studyingtypical arrival patterns during lunch hour. She records thenumber of arrivals for 40 randomly selected 15-minuteintervals over lunch hour, and obtains the following data: 7, 5, 2, 6, 2, 6, 6, 4, 6, 6, 7, 5, 2, 2, 8, 6, 6, 6, 1, 5, 9, 6, 2, 9,6,11, 2, 3, 7, 5, 6, 8, 4, 4, 4, 7, 5, 7, 5, 5 •^ Make frequency and relative frequency table •^ Draw histogram

Example: Call Center Data

-^ Financial firm call center •^ Calls handled by AVI within 60 seconds^ –^ October: 666^ –^ December: 523 •^ Avi Service Time Data

December Histogram

12010080 60 40 20 0 6 12 18 24 30 36 42 48 54 60

calling time Frequency

Frequency

Notes for Making Histogram

-^ Choose the number of classes sensibly^ –^ Too few classes: “skyscraper” graph^ –^ Too many: “pancake” graph •^ Sturge’s rule:^ –^ Choose number of classes k such that

log^2 n <k < log

2 n + where n is the sample size

-^ Intervals must be of equal width. •^ Areas of the bars are proportional to thefrequency.

Shapes of Distributions

-^ Graphs can help to determine shapes.^ –^ Modes: peaks of a distribution. -^ Unimodal: one peak •^ Bimodal: two peaks – Symmetric or skewed?

Shapes of Distributions

-^ Symmetric

:

-^ histogram in which the right half is a mirror image of the lefthalf. • Skewed to the right

:

-^ histogram in which the right tail is more stretched out thanthe left.(long tail to the right) • Skewed to the left

:

-^ histogram the left tail is more stretched out than theright.(long tail to the left) • Bell-shaped

: (a special case of symmetric distribtions)^ –^ A histogram looks like a bell.