Measures of central tendency:- The main measure of central tendency are the arithmetic mean, mode, median. The average (arithmetic mean) of a group of number is defined as the sum of the values divided by the number of values.
AVERAGE VALUE = Sumof values ÷ Numberof values
MEDIAN: When there are odd numbers of terms in a set, the median of a set is the middle of terms in a set, median of a set is the middle term.When all the terms in a set are listed in sequential order. When there are even number of terms in a set, the median of of a set is the average of two middle terms.
MODE:- Mode is the number that appears most frequently in a list of numbers.
MEASURES OF DISPERSION:- The main measures of dispersion are the range interquartile range and standard deviation.
First Quartile, Q is the median of all the numbers below the median. The second quartile, Q2 is the median of the entire data set. The third Quartile, Q3 is the median of all the numbers above the median. The interquartile range is the difference between Q3 and Q1/
The range and interquartile range are often displayed on a Box AND Whisker plot also called a Box Plot.
EXAMPLE:- For the following Set : 2,2,3,4,5,5,6,21
Calculate Q1, Q2, Q3 interquartile range plot box and whisker plot for the following data?
Ans:- = Median = Mean of 4th and 5th term= (4+s / 2) = 4.5 = Q
Q1 is the median of 5,5,6, and 21 , Which is 5.5.
Interquartile range :-Q3 – Q1 = 5.5 -2.5 = 3
RANGE = 21-2 = 19
QUES:-Which of the following set of data applies to this box and whisker plot?
Ans:-From this box – and – Whisker Plot, We can easily reckon that Q1 = – 3, Q2 =-1 and Q3 = 0
Median of ” -4,-4,-2,0,0,5” is -1 , Median of “ -4,-4,-2,-2,0,0,0,5” is -1
Q1 for Set n”-4,-4,-2,0,0,5”= Median ofd “-4,-4,-2”
Q1 for set “-4,-4,-2,-2,0,0,0,5” = Median of “-4,-4,-2,-2”
=(-4-2 / 2) = ( – 3 )
“-4,-4,-2,-2,0,0,0,5” is the only answer choice left. We can choose it without clicking if you are confident in your previous work.
Standard Deviation:- Standard deviation is a measure of how spread out a set of numbers is
For a set (h1 , h2 , h3 , _ _ _ _ n ), in Which x is the mean.
Calculate the following standard deviation for the data set 0,7,8,10 and 10?
Ans:-Mean = (0+7+8+10+10 / 5 ) = 7
Normal Distribution:- In the normal distributions, data points tend to cluster around the mean properties of normal distribution curve:-
1)The mean is at the center and the highest point on the curve and because the distribution is symmetrical about the mean, the mean is also the median and mode.
2)The standard deviation since it is a measure of dispersion determines the width of normal distributors. The greater the standard deviation, the wider and flatter the normal distribution curve is.
3)The probability of a data point falling within one standard deviation above the mean or below the mean is 34%
4)The probability of a data point falling between (mean -d ) and (mean -2d) is 13.5%
5)The probability of a data point falling between (mean +d) and (mean +2d) is 13.5%
NOTE: The values shown in the diagram above for various key areas of the curve are approximately likely to see slightly different value in Questions.
QUES:- A Food manufacturer produces energy bars that have a mean weight of 50grams. If given days production is 10,000 energy bars, how many of those bars would be expected to weight between 49.0 and 49.5 grams?
(Assume that the weight are normally distributed)
MEAN = 50 GRAM
68% of the bar weight between 49.5gm and 50.5 gm. Since 68% Corresponds to the area represented between (mean +d) and (mean -d). Therefore standard deviation= 0.5gm
13.5% of these bars weight between 49 and 49.5%.
(13.5/1000*10,000) = 1350
Ques:- How values among the 8000 homeowners of town X are normally distributed, with a standard deviation of $11,000 and a mean of $ 90,000.
|Quantity A||Quantity B|
| The number of homeowners in Town X , Whose home value is greater than $ 112,000
Number of homeowners in town X whose home value is greater than $ 112,000
2.5% of 8000
= 2.5 / 100 * 8000 = 200
Therefore , Quantity B is Greater.
Ques:- On a particular test whose score is distributed normally, the 2nd percentile is 1720, while the 84th percentile if 1,990. What score round to the nearest 10, most closely corresponds to the 16th percentile?
Mean -2x (deviation) = 1720
Mean + x(deviation) = 1990
3x(deviation) = 270
Mean = 1720+2 (90)= 1900
Mean – deviation = 1900-90= (1810)
Ques:- If a set of data consist of only the first ten positive multiples of 5, What is the interquartile range of the set?
Ans:- The firstten positive multiples of 5 are : 5,10,15,20,25,30,35,40,45,50.
Q1 (Median of [5,10,15,20,25] ) =15
Q3 (Median of [30,35,40,45,50] ) =40
INTERQUARTILE Range = Q3- Q1= 40-15 = (25)