NUMERICAL AND STATISTICAL METHODS Name: Aabha A TiwariDept: Computer EngineeringTopic: Statistical Methods

DATA ANALYSIS:Data analysis is a process of inspecting,transforming and modeling data with a goal to discover useful information and then correctly interpret it’s application. Ungrouped Data:The data obtained in original form are called raw data or

ungrouped data. Pictograms or Picture Diagram Bar Charts or Bar Diagram Presentation of

ungrouped data Pie Diagram Grouped Data:To put the data in a more condensed form, we make groups of

suitable size and so it is called grouped data. Histograms Frequency Polygon and frequency curve Presentation of

grouped data Ogive or Cumulative frequency distribution curve

PICTOGRAMS OR PICTURE DIAGRAMS:A pictogram or picture diagram represents the frequency of data as pictures or symbols. Each picture or symbol may represent one or more units of the data.Example:The following table shows the number of computers sold by a company for the months January to March. Construct a pictograph for the table.

Solution:

BAR CHARTS OR BAR DIAGRAMS:A Bar Chart is a graphical display of data using bars of different heights.

Vertical Bar Chart: Horizontal Bar Chart:

PIE DIAGRAMS:Pie diagram is a special chart that uses "pie slices" to show relative sizes of data.Example:

Solution:

GROUPED DATA:

Exclusive: Inclusive:

In this, the class intervals are 0 - 10, 10 - 20, 20 - 30. In this, we include lower limit but exclude upper limit.So, 10 - 20 means values from 10 and more but less than 20.20 - 30 would mean values from 20 and more but less than 30.

Here, also we arrange the data into different groups called class intervals, i.e., 0 - 10, 11 - 20, 21 - 30.0 to 10 means between 0 and 10 including 0 and 10.Here, 0 is the lower limit and 10 is the upper limit. 11 to 20 means between 11 and 20 including 11 and 20.Here, 11 is the lower limit and 20 is the upper limit.

HISTOGRAM:Histogram is a graphical display of data using bars of different heights.It is similar to bar chart but a histogram groups numbers into ranges.

FREQUENCY POLYGON:A frequency polygon is a way to show the information in a frequency table.Suppose an example:

Finding midpoints

The reason it’s called a polygon is because the line sort of forms a plane shape with the horizontal axis as one side of the shape

CAN BE ALSO FORMED USING HISTOGRAM:

If the class intervals are very small the frequency polygon assumes the form of a smooth curve which is called frequency curve.

OGIVE OR CUMULATIVE FREQUENCY DISTRIBUTION CURVE:The curve obtained by joining the co-ordinates of cumulative frequency(vertically) against upper class boundary(horizontally) is called an ogive. Ogives are used to visually represent how many values are below a certain upper class boundary.Example: Solution:

MEAN IN GENERAL TERMS:The mean is the average of the numbers i.e. a calculated "central" value of a set of numbers. To calculate: Just add up all the numbers, then divide by how many numbers there are.

Example: What is the mean of 2, 7 and 9?Add the numbers: 2 + 7 + 9 = 18Divide by how many numbers (i.e. we added 3 numbers): 18 ÷ 3 = 6So the Mean is 6Example:

MEAN:

_X = Σ f xm

The only one formula in solving the mean for grouped data is called midpoint method. The formula is:

_X = Σ f xm

n

n

_Where X = mean value xm = midpoint of each class or category

f = frequency in each class or category Σ f xm = summation of the product of f xm

MEAN:Example: Scores of 40 students in a science class consist of 60 items and they are

tabulated below:

X f Xm fXm10 – 14 5 12 6015 – 19 2 17 3420 – 24 3 22 6625 – 29 5 27 13530 – 34 2 32 6435 – 39 9 37 33340 – 44 6 42 25245 – 49 3 47 14150 - 54 5 52 260

n = 40 Σ f Xm = 1 345

_X = Σ f xm

n = 1 345 40 = 33.63

The mean performance of 40 students in science quiz is 33.63.

MEDIAN:The Median is the "middle" of a sorted list of numbers. Median is what divides the scores in the distribution into two equal parts.Fifty percent (50%) lies below the median value and 50% lies above the median value.

MODE:The mode is simply the number which appears most often. Finding the Mode:To find the mode, or modal value, first put the numbers in order, then count how many of each number. A number that appears most often is the mode.

MODE:It is classified as unimodal, bimodal, trimodal or mulitimodal.1)Unimodal is a distribution of scores that consists of only one mode.2)Bimodal is a distribution of scores that consists of two modes.3)Trimodal is a distribution of scores that consists of three modes.4)Multimode is a distribution that consisits of more than two modes.

MODE

Example: Scores of 10 students in Section A, Section B and Section C.

Scores of Section A

Scores of Section B

Scores of Section C

25 25 2524 24 2524 24 2520 20 2220 18 2120 18 2116 17 2112 10 1810 9 187 7 18

The score that appeared most in Section A is 20, hence, the mode of Section A is 20. There is only one mode, therefore, score distribution is called unimodal.The modes of Section B are 18 and 24, since both 18 and 24 appeared twice. There are two modes in Section B, hence, the distribution is a bimodal distribution.

The modes for Section C are 18,21 and 25.There are three modes for Section C,therefore it is called a trimodal or multimodal distribution.

PERCENTILE:The value below which a percentage of data falls.

To calculate percentile:The data needs to be in order.So here to calculate percentiles of height the data needs to be in height order (sorted by height). Similarly to calculate percentiles of age the data needs to be in age order.

DECILE:A related idea is Deciles (sounds like decimal and percentile together), which splits the data into 10% groups.•The 1st decile is the 10th percentile (the value that divides the data so that 10% is below it) •The 2nd decile is the 20th percentile (the value that divides the data so that 20% is below it)…etc

QUARTILES:Another related idea is Quartiles, which splits the data into quarters.

The Quartiles also divide the data into divisions of 25%:•Quartile 1 (Q1) can be called the 25th percentile•Quartile 2 (Q2) can be called the 50th percentile•Quartile 3 (Q3) can be called the 75th percentile

ESTIMATING PERCENTILE:

STANDARD DEVIATION AND VARIANCE:

The Standard Deviation is a measure of how spread out numbers are.

Its symbol is σ (the greek letter sigma)

The formula is easy: it is the square root of the Variance.

So now you ask, "What is the Variance?" Variance is the average of the squared differences from the Mean.

To calculate the variance follow these steps:•Work out the Mean. •Then for each number: subtract the Mean and square the result (the squared difference). •Then work out the average of those squared differences.

EXAMPLE:

Why we use squared mean?

Why do we need sample standard deviation?Suppose you want to know what the whole country thinks…you can’t

ask millions of people,so instead you ask maybe 1,000 people.This is what we did in our design engineering survey.So,this is the importance of sample standard deviation.

CORRELATION:The word Correlation is made of Co- meaning "together" and Relation.So,when two sets of data are linked together we say they have a Correlation.

Correlation can have a value: 1 is a perfect positive correlation. 0 is no correlation (the values don't seem linked at all). -1 is a perfect negative correlation.

Correlation is Positive when the values increase together. Correlation is Negative when one value decreases as the other increases.

This is how different correlation graph looks like.

We can easily see that warmer weather leads to more sales,the relationship is good but not perfect.

CORRELATION IS NOT CAUSATION: