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Lecture 2 – Grouped Data Calculation

1. Mean Median and Mode. 2. First Quantile

9 Data Analysis - 9.1 Mean Median

https://www.cimt.org.uk/projects/mepres/allgcse/bkb9.pdf

Finding the Mean Median

https://www.riosalado.edu/web/oer/wrkdev100-20011_inter_0000_v1/m5/pdf/m5_l1_mean_median_mode_practice_probs.pdf

Chapter # 03 Measures of Central Tendency

Mode. 5. Median. Arithmetic Mean or Simply Mean: “A value obtained by dividing the Using formula of direct method of arithmetic mean for grouped data:.

RS Aggarwal Solutions for Class 10 Maths Chapter 18- Mean

Mean Median

STATISTICS AND PROBABILITY

16-Apr-2018 (i) When the number of observations (n) is odd the median is the value of the ... Mode of ungrouped data can be determined by observation/.

Measures of Central Tendency

Mode is a measure that is less widely used compared to mean and median. There can be more than one type mode in a given data set. Computing Mode for Ungrouped

Exercise 18A Page No: 835

RS Aggarwal Solutions for Class 10 Maths Chapter 18 Mean. Median

15.1 Overview

In earlier classes, you have studied measures of central tendency such a s mean, mode, median of ungrouped and grouped data. In addition to these measures, we often need to calculate a second type of measure called a measure of dispersion which meas- ures the variation in the observations about the middle value- mean or median etc. This chapter is concerned with some important measures of dispersion suc h as mean deviation, variance, standard deviation etc., and finally analysis of frequency distributions.

15.1.1 Measures of dispersion

(a)RangeThe measure of dispersion which is easiest to understand and easiest to calculate is the range. Range is defined as: Range = Largest observation - Smallest observation (b)Mean Deviation (i) Mean deviation for ungrouped data:

For n observation x1, x2, ..., xn, the

mean deviation about their mean x is given by M.D ( x) =| |ix x n - (1)

Mean deviation about their median M is given by

M.D (M) =| M|ix

n- (2) (ii) Mean deviation for discrete frequency distribution Let the given data consist of discrete observations x1, x2, ... , xn occurring with frequencies f1, f2, ... , fn, respectively. In this caseChapter 15

STATISTICS

M.D (x) =

Ni ii i

i f x x f x x f = (3)

M.D (M) =

| M| Ni i f x- (4) where N = if . (iii) Mean deviation for continuous frequency distribution (Grouped d ata). M.D ( x) = Ni i f x x- (5)

M.D (M) =

| M| Ni i f x- (6) where xi are the midpoints of the classes, x and M are, respectively, the mean and median of the distribution. (c)Variance : Let x1, x2, ..., xn be n observations with xas the mean. The variance, denoted by σ2, is given by 2 =

21( )ix xn- (7)

(d)Standard Deviation: If σ2 is the variance, then σ, is called the standard deviation, is given by

21( )ix xn- (8)

(e)Standard deviation for a discrete frequency distribution is given by

21( )Ni if x x- (9)

where fi's are the frequencies of xi' s and N = 1n i i f (f)Standard deviation of a continuous frequency distribution (grouped data) is given bySTATISTICS 271

272 EXEMPLAR PROBLEMS - MATHEMATICSσ =21( )Ni if x x- (10)

where xi are the midpoints of the classes and fi their respective frequencies.

Formula (10) is same as

1 N ( )

22Ni i i if x f x- (11)

(g)Another formula for standard deviation : x = N h ( )

22Ni ii if y f y- (12)

where h is the width of class intervals and yi = Aix h -and A is the assumed mean.

15.1.2 Coefficient of variation It is sometimes useful to describe variability by

expressing the standard deviation as a proportion of mean, usually a per centage. The formula for it as a percentage is

Coefficient of variation =

Standard deviation100Mean×15.2 Solved Examples

Short Answer Type

Example 1 Find the mean deviation about the mean of the following data:

Size (x):13

5 7

91113 15

Frequency (f):3 3 4147 4 3 4

Solution Mean =

x =3 9 20 98 63 44 39 60 42i i
if x f+ + + + + + +=

336842=M.D. (

x) =| |3(7) 3(5) 4(3) 14(1) 7(1) 4(3) 3(5) 4(7) 42i i
if x x f-

STATISTICS 273

=21 15 12 14 7 12 15 28 62 4221
+ + + + + + += = 2.95

Example 2

Find the variance and standard deviation for the following data:

57, 64, 43, 67, 49, 59, 44, 47, 61, 59

Solution Mean (

x) =57 64 43 67 49 59 61 59 44 47 550551010 + + + + + + + + += =Variance (σ2) =

2( )ix x

2 2 2 2 2 2 2 2 2 22 9 12 12 6 4 6 4 11 8

66266.210=Standard deviation (σ) =

266.2 8.13σ = =Example 3 Show that the two formulae for the standard deviation of ungrouped data.

2( )ix x

n-σ = and 2

2ixxnσ′ = - are equivalent.

Solution We have

2( )ix x- =

22( 2 )i ix x x x- + =

222iix x x x+ - + =

( )2221iix x x x- + =

222 ( )ix x nx n x- + =

2 2quotesdbs_dbs7.pdfusesText_5

[PDF] STATISTICS such as mean mode

15.1 Overview

15.1.1 Measures of dispersion

For n observation x1, x2, ..., xn, the

Mean deviation about their median M is given by

M.D (M) =| M|ix

STATISTICS

M.D (x) =

Ni ii i

M.D (M) =

M.D (M) =

21( )ix xn- (7)

21( )ix xn- (8)

21( )Ni if x x- (9)

272 EXEMPLAR PROBLEMS - MATHEMATICSσ =21( )Ni if x x- (10)

Formula (10) is same as

22Ni i i if x f x- (11)

22Ni ii if y f y- (12)

15.1.2 Coefficient of variation It is sometimes useful to describe variability by

Coefficient of variation =

Short Answer Type

Size (x):13

91113 15

Frequency (f):3 3 4147 4 3 4

Solution Mean =

336842=M.D. (

STATISTICS 273

Example 2

57, 64, 43, 67, 49, 59, 44, 47, 61, 59

Solution Mean (

2( )ix x

2 2 2 2 2 2 2 2 2 22 9 12 12 6 4 6 4 11 8

66266.210=Standard deviation (σ) =

266.2 8.13σ = =Example 3 Show that the two formulae for the standard deviation of ungrouped data.

2( )ix x

2ixxnσ′ = - are equivalent.

Solution We have

2( )ix x- =

22( 2 )i ix x x x- + =

222iix x x x+ - + =

222 ( )ix x nx n x- + =