Statistics in research

STATISTICS IN
RESEARCH (PART – 1
DESCRIPTIVE)

Role of statistics in research
 Designing research
 Analyzing data
 Draw conclusion about research

Two major areas of statistics
 Descriptive statistics
 It concern with development of certain indices from the raw
data.
 It summarizes collected/ classified data.
 Inferential statistics
 It adopts the process of generalization from small groups
(i.e., samples) to population.
 It also known as sampling statistics.
 It concern with two major problems
 Estimation of population parameters.
 Testing of statistical hypothesis.

Descriptive statistics
 Measure of central tendency (Statistical
Average)
 Measure of dispersion
 Measure of asymmetry (Skewness)
 Measure of relationship
 Other measures

Measure of Central tendency
 It also known as statistical average.
 Mean, Median and Mode are the popular
averages.
 Geometric and Harmonic mean are also
sometime used.

Mean
 It also known as arithmetic average.
 The mean is found by adding all the values in
the set, then dividing the sum by the number of
values.

Example
 Find the mean of following numbers 8, 9, 10,
10, 10, 11, 11, 11, 12, 13
The mean is the usual average:
 n= 10
 (8 + 9 + 10 + 10 + 10 + 11 + 11 + 11 + 12 + 13)
÷ 10 = 105 ÷ 10 = 10.5

Weighted Mean
 The weighted average formula is used to
calculate the average value of a particular set
of numbers with different levels of weight.

Example
 Final marks has distribution of internal marks,
Mid semester marks and End semester marks.
Each one carries following weightage
Internal Mid-Sem End-Sem
30 20 50
Shakthi got 80 marks in internal, 75 marks in
Mid sem and 89 marks in End sem. Then what
is his final mark? Ans = 83.5

Frequency distribution Mean
Parkin
g
Spaces
Freque
ncy
1 15
2 27
3 8
Isabella went up and down the street to find out how many parking
spaces each house had. Here are her results: What is the mean
number of Parking Spaces?
Mean = 15×1 + 27×2 + 8×3 + 5×4
15+54+24+20 /15+27+8+5
Mean = 2.05

Median
 Median is the value of the middle item of
series when it is arranged in ascending or
descending order of magnitude.
 It divides the series into two halves; in one half
all items are less than median, whereas in the
other half all items have values higher than
median

Example
 21, 18, 24, 19, 27
 Arrange the number in ascending order
 18, 19, 21, 24, 27 (21 is the median)
 If there are two middle numbers,
 18, 19, 21, 25, 27, 28
 (21 + 25 )/2 = 23

Mode
 The number that appears most frequently in a set
of numbers.
 It useful in the studies of popular. (Popular shoe
size, popular cap, most demanded product etc.,)
 Arrange the numbers in order from least to
greatest.
 21, 18, 24, 19, 18
 Find the number that is repeated the most.
 18, 18, 19, 21, 24 ( 18 is the mode).

Geometric Mean
 It is defined as the nth root of the product of the values of n times in
a given series.
 The most frequently used application of this average is in the
determination of average percent of change.

Example
 What is the geometric mean of 4,6 and 9 ?
Ans: 6
 Your investment earns 20% during the first year, but
then realizes a loss of 10% in year 2, and another
10% in year 3
 Calculate a growth factor for each year. (1+.2) for Year 1, (1-
.1) for year 2 and (1-.1) for Year 3
 Multiply the 3 growth factors and take the 3rd root.
 Thus, geometric mean = 0.990578-1=-0.009422.
 So your investment losing roughly .9 percent of every year.

Harmonic Mean
 It is defined as reciprocal of the average of
reciprocal of values of items of a series.
 It has limited application particularly in time
and rate are involved.
 It gives largest weight to smallest value and
smallest weight to largest value.

Harmonic Mean
What is the Harmonic mean of 4,5 and 10?
Ans: 5.45

Example
 For example, suppose that you have four 10 km
segments to your automobile trip. You drive your
car:
 100 km/hr for the first 10 km
 110 km/hr for the second 10 km
 90 km/hr for the third 10 km
 120 km/hr for the fourth 10 km.
 Ans: 103.8 km/hr.

Measure of Dispersion
 Average can’t reveal the entire story of study. It
fail to give ideas about data which distributed
around average.
 Measures of dispersion measure how spread out
a set of data is.
 Important measure of dispersion are
 Range
 Mean Deviation
 Standard deviation

Range
 It is difference between extreme values.
 Range = Highest value – Lowest value
 What is the range for following data?
 2, 3, 1, 1, 0, 5, 3, 1, 2, 7, 4, 0, 2, 1, 2, 1, 6, 3, 2, 0,
0, 7, 4, 2, 1, 1, 2, 1, 3, 5, 12
 Range = 12
 Limitation: Its value is never stable, being based
on only two values of the variable.

Mean Deviation
 It is the average of difference of the values of items
from some average of the series
 Find the mean deviation of 3, 6, 6, 7, 8, 11, 15, 16
Find the mean
Find the absolute distance
from the mean.
Find mean of those
Distances.
Ans= 3.75

Coefficient of mean deviation
 When mean deviation is divided by the
average used in finding out the mean deviation
itself, the resulting quantity is described as the
coefficient of mean deviation.
 Coefficient of mean deviation is a relative
measure of dispersion and is comparable to
similar measure of other series.

Standard deviation
 It is most widely used measure of dispersion of
a series and is commonly denoted by the
symbol ‘ 𝜎’ (pronounced as sigma).

Standard deviation
 Find the standard deviation of 53,61,49,67,
55,63.
Steps:
 Find the mean (58)
 Find the deviation from mean, square it and sum it
(230)
 Divide the above answer by sample size (38.333)

Other terms related to 𝜎
 Variance : Square the standard deviation
 Coefficient of standard deviation: Divide the
standard deviation by Mean of that data.
 Coefficient of variation: Multiply the coefficient
of standard deviation with 100.

Distribution
 The pattern of outcomes of a variable; it tells
us what values the variable takes and how
often it takes these values.
 The distribution of data can be
find with help of histogram.
 Histogram is bar chart.

Steps in making Distribution
 Choose the classes by dividing the range of data
into classes of equal width (individuals fit into one
class).
 Count the individuals in each class (this is the
height of the bar).
 Draw the histogram:
The horizontal axis is marked off into equal class
widths.
The vertical axis contains the scale of counts

Histogram
 The number of days of Maria’s last 15
vacations are listed below. Use the data to
make a frequency table with intervals.
4, 8, 6, 7, 5, 4, 10, 6, 7, 14, 12, 8, 10, 15, 12.
 Step 1: Identify the least and greatest values.
 Step 2: Divide the data into equal intervals.

Histogram
 Step 3: List the intervals in the first column of
the table. Count the number of data values in
each interval and list the count in the last
column. Give the table a title.

Normal Distribution
 The distribution of data happens to be
perfectly symmetrical.
 It is perfectly bell shaped curve in which case
the value of mean 𝑋 = median M = mode Z.

Skewness
 if the curve is distorted (whether on the right
side or on the left side), we have asymmetrical
distribution which indicates that there is
skewness.
 If the curve is distorted on the right side, we
have positive skewness.
 If the curve is distorted on the left side, we
have negative skewness.

Measure of Skewness
 Skewness = 𝑋 - Z or 3 ( 𝑋 - M)
 Coefficient of Skewness =
( 𝑋− 𝑍)
𝜎
 If skewness value is positive/ negative/ zero,
then data are positively/negatively
skewed/symmetry.

Kurtosis
 Kurtosis is the measure of flat-toppedness or
peakedness of a curve.
 Normal curve is MesoKurtic..
 Positive kurtosis is leptokurtic.
 Negative kurtosis is platykurtic

Measure of Relationship
 Univariate Analysis: The analysis is carried out
with the description of a single variable.
 Bivariate Analysis: The analysis of two
variables simultaneously.
 Multivariate Analysis: The analysis of multiple
variables simultaneously.

Measure of Relationship
 Correlation
 The word Correlation is made of Co- (meaning "together"),
and Relation.
 It answers that Does there exist association or correlation between
the two (or more) variables ?
 Regression
 It answers that:
 Is there any cause and effect relationship between the two variables
in case of the bivariate population or between one variable on one
side and two or more variables on the other side in case of
multivariate population? If yes, of what degree and in which direction?

Karl Pearson’s coefficient of
correlation
 It is simple correlation and most widely used
method. Denotes by r.
 It is also known as the product moment
correlation coefficient.
 The value of r lies between ±1.

r value and interpretation
We can also say that for a unit change in independent variable, if there happens
to be a constant change in the dependent variable in the same direction, then
correlation will be termed as perfect positive.

Karl Pearson Correlation coefficient
formula
r
If substitute the value of sigma and derive, you will get below
equation

Example
 A researcher want to know the relation between
advertisement expenditure and total sales. He took a sample
data of 7 companies for one year.

Regression
 It is the study of the relationship between
variables.
 It also used for prediction of dependent
variable.
 Regression types
 Simple Regression: single explanatory/independent
variable

Regression Analysis
 Linear Regression: Straight-line relationship.
 Non-linear: Implies curved relationships.
 Regression is nothing but try to find out the equation
of line/curve.
Linear Non linear
y=mx+b

Linear Regression Equation
Unknown parameter a and b can be calculated by below
formulam = b = slope of line, a= c = intercept
Above formula gives best fit line by using least square method.
b = Σ [ (𝑥𝑖 - 𝑥)(𝑦𝑖 - 𝑦) ] / Σ [ (𝑥𝑖 − 𝑥)2
]
a = 𝑦 - b * 𝑥

Example for regression
 In the table below, the X column shows scores
on the aptitude test. Similarly, the Y column
shows statistics grades.
Student X Y
1
2
3
4
5
95
85
80
70
60
85
95
70
65
70

Regression
S.no xi yi (xi - x) (yi - y) (xi -
x)2
(yi -
y)2
(xi - x)(yi - y)
1 95 85 17 8 289 64 136
2 85 95 7 18 49 324 126
3 80 70 2 -7 4 49 -14
4 70 65 -8 -12 64 144 96
5 60 70 -18 -7 324 49 126
Sum 390 385 730 630 470
Mean 78 77
b = 470/730 = 0.644 a = 26.768 ŷ = 26.768 + 0.644x

Standard error
 It provide an overall measure of how well the model
fits the data.
 It represents the average distance that the
observed values fall from the regression line.
 Smaller values are better because it indicates that
the observations are closer to the fitted line.

Standard error
Y 𝑌 (𝑌 − 𝑌) (𝑌 − 𝑌)2
85 81.508 3.492 12.19406
95 87.948 7.052 49.7307
70 71.848 -1.848 3.415104
65 68.628 -3.628 13.16238
70 71.848 -1.848 3.415104
Sum 81.91736
Standard error = 4.0475

Multiple regression
Multicollinearity: Independent variables in multiple regression are highly
correlated.

Partial Correlation
 In simple correlation, we measure the strength of the linear
relationship between two variables, without taking into
consideration the fact that both these variables may be
influenced by a third variable.
 For Ex: when we study the correlation between price and demand,
we completely ignore the effect of other factors like money supply,
import and exports etc. which definitely have a bearing on the price.
 The correlation co-efficient between two variables X1 and X2,
studied partially after eliminating the influence of the third
variable X3 from both of them, is the partial correlation co-
efficient.

Other Measures
 Index number
 Indicator of average percentage change in a series of figures
where one figure (called the base) is assigned an arbitrary
value of 100, and other figures are adjusted in proportion to
the base.
 Time series analysis
 Unlike the analyses of random samples of observations that
are discussed in the context of most other statistics, the
analysis of time series is based on the assumption that
successive values in the data file represent consecutive
measurements taken at equally spaced time intervals.

Statistics in research

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