quant hw

intro

Control charts

To create a control chart

are used to monitor and improve processes

collect subgroups (samples) of process output over time

data is collected over time

calculate sample statistics for each subgroup

past performances are use to predict future outcomes

plot the statistics over time

add control limits (ususally within +- 3 standard deviations of the statistic of measure

Phase 1 control charts

Goals are to find patterns over time and points that fall outside limits

analyzed at the start of a process to determine where improvements are needed

Phase 2 control charts

analyed after improvements have been imposed

Causes of variation

special and common

special causes are correctable without changing the process

common cause variation is inherit in the process (random)

common cause variation (no points outside limits) - stable process - in control - predictable

A rule of thumb is 8 or more consecutive points

above or below the center line may indicate a trend

special cause variation (point outside limit) - out of control process

unpredictable

p chart

used for categorical variables
					ni = Number of observations
						to find p-bar (mean) divide sum of pi by the number of observations
number of Days	10
subgroup size	100									Generate the plot
					p-bar	(p-bar(1-p-bar))/ni	UCL	LCL		pi	LCL	UCL
	Day	defects	proportion (pi) defects per day		0.147	0.00125391	0.2532317749	0.0407682251		0.12	0.041	0.253
	1	12	0.12							0.14	0.041	0.253
	2	14	0.14							0.1	0.041	0.253
	3	10	0.1							0.18	0.041	0.253
	4	18	0.18							0.21	0.041	0.253
	5	21	0.21							0.14	0.041	0.253
	6	14	0.14							0.15	0.041	0.253
	7	15	0.15							0.12	0.041	0.253
	8	12	0.12							0.15	0.041	0.253
	9	15	0.15							0.16	0.041	0.253
	10	16	0.16
sum of pi			1.47
p-bar (mean)			0.147

pi 0.12 0.14000000000000001 0.1 0.18 0.21 0.14000000000000001 0.15 0.12 0.15 0.16 LCL 4.0768225092489396E-2 4.0768225092489396E-2 4.0768225092489396E-2 4.0768225092489396E-2 4.0768225092489396E-2 4.0768225092489396E-2 4.0768225092489396E-2 4.0768225092489396E-2 4.0768225092489396E-2 4.0768225092489396E-2 UCL 0.25323177490751059 0.25323177490751059 0.25323177490751059 0.25323177490751059 0.25323177490751059 0.25323177490751059 0.25323177490751059 0.25323177490751059 0.25323177490751059 0.25323177490751059

Number of Days

Probability

c chart

monitors errors in an area of opportunity (space, time, etc)
number of areas(time) sampled (n)			10										ci is the number of errors in area i
						Graph the chart
	Day	errors (ci)				Ci	UCL	LCL
	1	7
	2	3
	3	6
	4	3
	5	4
	6	5
	7	3
	8	5											SQRT is the square root feature in Excel
	9	2
	10	0
sum ci
c-bar (Mean)

r chart

Use when observation size is 10 or less
	Is the process range in control?
	number of days collection occurs (k)			10
	observation size (use in control constants table)			5					Graph the process
										From Control table
	Day	5 delivery times collected each day (Xi)					Ri		R-bar	D3	D4	UCL	LCL
	1	6.7	11.7	9.7	7.5	7.8
	2	7.6	11.4	9	8.4	9.2
	3	9.5	8.9	9.9	8.7	10.7
	4	11	9.9	11.3	11.6	8.5
	5	8.3	8.4	9.7	9.8	7.1								D3 and D4 are control group constants
	6	9.4	9.3	8.2	7.1	6.1									they represent relations between variation and mean for a given sample size
	7	10	10.7	9	8.2	11									values are found in tables
	8	9.5	10.5	7	8.6	10.1
	9	7.8	9	12	9.1	11.7
	10	9.9	10.1	8.9	9.6	7.1
sum Ri
R-bar