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What is a Histogram?Histogram is a visual tool for presenting variable data . Itorganises data to describe the process performance.Additionally histogram shows the amount and pattern of thevariation from the process.Histogram offers a snapshot in time of the process performance.
Why do We Get Variation?Variation is essentially law of nature.Output quality characteristics depends upon the input parameters.It is impossible to keep input parameters constant. There will bealways variation in the input parameters. Since there is variation inthe input parameters, there is also variation in the outputcharacteristics
Law of NatureIn nature there is always variation. Take case measurement of thefollowing: height of adult male in a city. weight of 15 years old boy in a town. weight of bars 5 meter long 25 mm dia. volume in 300 cc soft drink bottle. number of minutes required to fill an invoice.
Case when Data Does Not Show VariationThere could be two reasons when data do not show variation:a) Measuring devices are insensitive to spot variation.b) Too much rounding off the data while recording.
Insensitive Measuring DeviceIf the measuring device is not sensitive, enough to respond tosmall changes in value of the quality characteristics, variation willnot be reflected in the data. For example:Weighing gold chains by using weighing scale used forvegetables.
Too Much Rounding Off During Recording It could also be possible that too much rounding off might have been carried while recording the measurements. This normally happens when the column in data recording sheet is not wide enough to record all the decimal places of measurements. Because of paucity of the space, workmen round off observations on their own.
Definition of HistogramA histogram is a graphical summary of variation in aset of data.The pictorial nature of the histogram enables us to seepatterns that are difficult to see in a table of numbers.
Basic Elements for Construction of Histogram For constructing the histogram we need to know the following: Lowest value of the data set Highest value of the data set Approximate number of cells histogram have Cell width Lower cell boundary of first cell
Finding Lowest & Largest Value in Data Set If the number of observations in the data set is small, then finding smallest and largest value is not a problem. However, if the number of observations is large, then we require an easier way to get smallest value and largest value in the data set. This can be achieved by grouping the data in rows, columns and then scanning.
Organizing Data in Rows & ColumnsStep - 1Organise the data in a group of 5 or 10 1 2 3 4 5 3.56 3.46 3.48 3.42 3.43 3.48 3.56 3.50 3.52 3.47 3.48 3.46 3.50 3.56 3.38 3.41 3.37 3.49 3.45 3.44 3.50 3.49 3.46 3.46 3.42
Construction of HistogramStep - 2Generate 2 more columns to record Smallest value in each row in column ‘S’ Largest value in each row in column ‘L’
Construction of HistogramStep-3 Scan column ‘S’ to find smallest value in that column, S. S is overall smallest value in the data set. Scan column ‘L’ to find largest value in that column, L. L is overall largest value in the data set
Range of the Data SetStep-4Find range of the dataRange of data = Largest value - smallest value In our case Range R = L - S = 3.56 - 3.37 = 0.19
Initial Number of Cells in HistogramStep-5 Decide the initial number of cells, say K, a histogram shall have. Number of cells a histogram can have, depends upon the number of observations N, histogram is representing. There are three methods to decide initial number of cells. Note: The number of cells, K initially chosen may change when histogram is finally made
Table for Choosing Number of CellsMethod No. 1 Number of observation Number of cells (N) (K) Under 50 5 to 7 50 - 100 6 to 10 101 - 250 7 to 12 More than 250 10 to 20
Alternative Methods for Deciding No. of CellsMethod No. 2Number of cells, K = 1 + 2.33 Log 10 NMethod No 3Number of cells, K = N
Temporary Cell WidthStep-6Find temporary cell width, TCW Range (R) TCW = Number of cells chosen (K) 0.19 = 7 = 0.0271423
Rounding of Temporary Cell WidthTemporary cell width, TCW needs rounding off. For ease of plotting For getting distinct cell boundary
Construction of HistogramStep - 6Round off TCW to get class width Rounding off of TCW, should be in the multiple of 1 or 3 or 5 of least count. The multiple should be nearer to TCW
Least Count of the Data 1 2 3 4 53.56 3.46 3.48 3.42 3.433.48 3.56 3.50 3.52 3.473.48 3.46 3.50 3.56 3.383.41 3.37 3.49 3.45 3.443.50 3.49 3.46 3.46 3.42 Least count of the data is 0.01
Procedure for Getting Class WidthIn our case least count of the data,LC is 0.01and TCW = 0.0271428If multiple factor, M is 1 then we haveM × LC = 1 x 0.01 = 0.01This multiple is not nearer to TCWIf multiple factor is 3 then we haveM x LC = 3 x 0.01 = 0.03This multiple is nearer to TCWHence class width, CW = 0.03
Class BoundariesStep - 7 Determine class boundaries Class boundaries are necessary for making tally sheet. Frequency obtained in tally sheet is utilised for making histogram. Class boundaries should be distinct
Distinct Class BoundariesDistinct class boundaries are the one, on which no individual datalies.With the distinct class boundary the data will enter in a particularcell only.
Nomenclature of Cell BoundariesLet LCB(1), LCB(2), … are the lower cell boundaries of cell no.1,cell No. 2…. respectively.Let UCB(1), UCB(2), … are the upper cell boundaries of cell no.1,cell No. 2…. respectively.
Elements of Histogram Upper Lower cell boundary cell boundary of cell no. 2 of cell no. 2 Upper Lower cell boundary cell boundary of cell no. 1 of cell no. 3 Cell No. 2 Lower Cell Uppercell boundary Cell No. 3 cell boundary of cell no. 1 No. 1 of cell no. 3 CW CW CW Continuous Scale
Calculation of Cell BoundariesIf we know the lower cell boundary of cell No.1, LCB(1), and classwidth, CW we can find other cell boundaries as follows: UCB(1) = LCB(1) + CW LCB(2) = UCB(1) UCB(2) = LCB(2) + CW LCB(3) = UCB(2) and so on
Getting Lower Cell Boundary of Cell No.1 Choose a starting value A, which is slightly lower or equal to smallest value, S. Value of S in our case is 3.37 We can take A = 3.37 LCB = A - ( CW / 2 ) = 3.37 - ( 0.03 / 2 ) = 3.355
Getting Cell BoundariesUCB(1) = LCB(1) + CW = 3.355 + 0.03 = 3.385LCB(2) = UCB (1) = 3.385UCB(2) = LCB(2) + CW = 3.385 + 0.03 = 3.415Continue finding cell boundaries, till a particular upper cell boundaryis greater than the largest value of data set.
Filling of Frequency ColumnCount the number of tally marks in each cell andenter the count in ‘Frequency’ column Mid Tally SN Cell Boundary Frequency Value Marks 1 3.355 - 3.385 3.37 2 2 3.385 - 3.415 3.40 2 3 3.415 - 3.444 3.43 3 4 3.445 - 3.475 3.46 4 5 3.475 - 3.505 3.49 8 6 3.505 – 3.535 3.52 4 7 3.535 - 3.565 3.55 2
Drawing Histogram 9 8 Label vertical axis from zero to a multiple of 1, 2 or 5 to accommodate the largest frequency 7Frequency 6 5 4 3 Label horizontal axis with mid values of the cells, and indicate the dimension of quality characteristics 2 1 0 3.37 3.40 3.43 3.46 3.49 3.52 3.55 mm
Drawing Histogram 9 8 7Frequency 6 5 Leave one cell 4 width space from 3 vertical axis 2 1 0 3.37 3.40 3.43 3.46 3.49 3.52 3.55 mm
Drawing HistogramDraw bars to represent frequency in each cell. Height of bars isequal to number of data in each cell.Title the chart.Indicate total number of observations
Design Tolerance VS Process Spread LSL USL 16 Design Tolerance 14 Process SpreadFrequency 12 10 8 6 4 2 0 47 48 49 50 51 52 53 54 kg
Assessing Process CapabilityProcess capability is a comparison between design tolerance andspread of the process.Whenever design tolerance is more than process spread, then theprocess is capable.Whenever design tolerance is less than the spread of the process,then the process is not capable.
Assessing Process CapabilityLSL USL47 48 49 50 51 52 53 54 kg Process is not capable
Assessing Process Capability USLLSL47 48 49 50 51 52 53 54 kg Process is just capable
Assessing Process Capability LSL USL46 47 48 49 50 51 52 53 54 55 kg Process is capable
Assessing Process CapabilityLSL USL 47 48 49 50 51 52 53 54 kg At the moment process is not capable