Assignment Code: BPCC-104 / Asst /TMA /2021-22

Course Code: BPCC-104

Assignment Name: Statistical Methods for Psychological Research- I

Year: 2021-2022

Verification Status: Verified by Professor

NOTE: All assignments are compulsory.

Instructions:

1.Have a title page. Include details like Name, Enrolment number, Email id, Regional Centre,

Study Centre, Programme Title and code, Course title and code and Tutorial code.

2.Use A4 size paper for the tutorial (ruled/ bank).

3.For making tables, blank pages can be used and tables/ graphs (if any) to be drawn in pencil.

4. Content should not be plagiarised.

Part A

Assignment One

Answer the following questions in about 500 words each (wherever applicable).

Each question carries 20 marks. 2 x 20 = 40

Q1. Define Statistics and discuss the scales of measurement.

Ans) Statistics

1. Population: This term can be used to describe individuals, things, elements, animals, or even reactions that exhibit a distinctive pattern of traits. It can also be defined as a group of people, things, things made of animals, reactions, and things that the researcher wants to study. If a researcher wants to conduct a study on teenagers in New Delhi, then all of the teenagers in New Delhi will be the population. There are two types of populations: finite and infinite. The number of students in a school who have failed mathematics is an illustration of a finite population. The number of stars in the sky is another illustration of infinite populations.

2. Sample: The group of people who take part in the research is a straightforward definition of a sample. If we use the adolescents in New Delhi mentioned above as an example, it would be impossible for the researcher to reach out to and collect data from every adolescent in New Delhi. So, a sample from that population will be taken by the researcher.

3. Parameter :A parameter is a value that offers details about the population under investigation in the study. It can be characterised as "a population measure that refers to the indices of a central value, dispersion, correlation, and so on of all the population individuals." For instance, the mean weight of infants born in India in a given year can be used as a parameter because it indicates the average weight of all infants born in India in that year. Even though getting an exact parameter is not always simple, every parameter has a statistic.

Scales of Measurement

In the process of measuring, numbers are meaningfully assigned to observations. The characteristics of the quantification of the observations can vary. One kilogramme of wheat, for instance, equals half of two kilogrammes. On the other hand, grades can be given to students based on how well they perform in math. For instance, a student who placed first might have earned 95 marks, whereas a student who placed second might have earned 80 marks, and a student who placed third might have earned 79 marks. As can be seen, the numerical characteristics of the two examples vary.

1. Nominal Scale: It is possible to measure variables with a nominal scale that are both qualitative and exclusive in nature. such as gender, religion, and so forth. The word nominal comes from the Latin word "nominalis," which has to do with names. Despite the fact that these variables are qualitative in nature, numbers can be assigned to them.

2. Ordinal Scale: The data can be ranked according to how much less or more, how high, or low, how good, or bad they are, and so on when using an ordinal scale. Thus, the data is ranked according to its size.

3. nterval Scale: The most typical scale to measure psychological variables is the interval scale. As the categories can be ranked and are exclusive as well, these scales are comparable to the ordinal scale in that the degree of difference between two participants is the same.

Q2. What is nonlinear correlation? Compute Spearman’s rho for the following data:

Ans) Nonlinear Correlations

As opposed to linear relationship, in nonlinear relationship. The relationship between two given variables is not denoted by a straight line. Correlation is said to be linear if the ratio of change is constant. When the amount of output in a factory is doubled by doubling the number of workers, this is an example of linear correlation.

Spearman's Rho Calculator

A non-parametric test called Spearman's Rho is used to assess how strongly two variables are related; a value of r = 1 denotes a perfect positive correlation, while a value of r = -1 denotes a perfect negative correlation. So, for instance, you could use this test to determine whether a correlation exists between a person's height and shoe size.

Requirements

1. zzScale of measurement must be ordinal or interval, ratio

2. Data must be in the form of matched pairs

3. The association must be monotonic i.e., variables increase in value together, or one increases while the other decreases

Equation

  rs =1-(6∑D^2 )/(N^(3 )-N)

Assignment Two

Answer the following questions in about 100 words each (wherever applicable).

Each question carries 5 marks. 6 x 5 = 30

Q3. Explain classification and tabulation of qualitative and quantitative data.

Ans) Both qualitative and quantitative data are possible. Measures of types, qualitative data are identified by a name, a symbol, or a numerical code. They are information types with characteristics that cannot be measured. Qualitative data are information about categorical variables, to put it simply. Your skin's radiance, eye colour, hair texture, palm softness, and other physical characteristics are some examples of qualitative data.

Classification

Data classification is a technique for grouping data for the most effective and efficient use possible. Vital information is simple to locate and retrieve whenever needed with the help of a well-planned data classification system.

Tabulation

Inserting classified data into tabular form is the process of tabulating. A table is a symmetrical set of rows and columns that contains statistical data. Columns are arranged vertically, whereas rows are arranged horizontally.

Q4. Compute mean, medina and mode for the following data.

Ans) Median

The middle value in a data set is called the median. Put all of your numbers in ascending order before calculating it. The next step is to locate the middle number on your list if you have an odd number of integers. The median or middle number in this instance is 15:

3, 9, 15, 17, 44

A further step or two are needed to calculate the median if the number of data points is even. Find the middle two numbers in your list first. After adding them up, divide the result by two. The outcome is the median value. The two middle numbers in this illustration are 8 and 12:

3, 6, 8, 12, 17, 44

Written out, the calculation would look like this:

(8 + 12) / 2 = 20 / 2 = 10

In this instance, the median is 10.

Mode

The most frequent integers in a list of numbers are referred to as the mode in statistics. The mode talks about the frequency of occurrence, unlike the median and mean. Depending on the data set, there might be multiple modes or none at all. Let us take the following list of numbers as an illustration:

3, 3, 8, 9, 15, 15, 15, 17, 17, 27, 40, 44, 44

Since 15 is the most frequent integer in this instance, it is considered to be the mode. However, if your list had one fewer 15, you would have four modes instead: 3, 15, 17, and 44.

Q5. Describe frequency polygon with the help of a diagram.

Ans) Frequency polygon is the name of a line graph used to display a frequency distribution. The upper base of the histogram's midpoints can be directly drawn through to create a frequency polygon, or a straight line can be drawn through them to create a frequency polygon. The following steps are taken when drawing a frequency polygon:

Step 1

The frequency polygon is based on the frequency distribution, as is well known. Before drawing a frequency polygon, two additional class intervals—one below and one above—are added in the case of frequency polygons as well.

Step 2

Midpoints are calculated for each of the class intervals.

Step 3

The frequency polygon has an x axis and y axis, just like every other graph. The frequencies will be represented on the y axis, and the midpoints will be plotted on the x axis.

Step 4

The class intervals' corresponding frequencies are then plotted based on the midpoints specified on the x axis.

Step 5

These points are then joined to form a line.

Q6. Compute standard deviation for the following data.

Ans) Standard Deviation, σ: 9.1809585556193

Count, N: 10

Sum, Σx: 271

Mean, μ: 27.1

Variance, σ2: 84.29

Steps:

= σ2 = Σ(xi - μ)2 / N

= (34 - 27.1)2 + ... + (23 - 27.1)2 / 10

= 842.9 / 10

= 84.29

σ = √84.29

= 9.1809585556193

Q7. Elucidate the concept of variability with a focus on absolute and relative dispersion.

Ans) The deviation of individual scores in a given sample from the mean and median is expressed quantitatively and is estimated by measures of dispersion. As a result, the numerical measures of variability converge on or deviate from a central value. It is critical to understand both the magnitude and the direction of variation when measuring dispersion. We take into account the range, mean deviation, standard deviation, etc. in the first scenario. In the latter scenario, we take into account the range coefficient, mean deviation coefficient, variation coefficient, etc. As a result, there are two major categories of dispersion or variability measures. They are the absolute and relative measures of dispersion, respectively. Standard deviation, a measurement of deviation from the mean, is frequently used to describe absolute dispersion.

Q8. Explain the concept and nature of normal probability distribution with help of suitable diagram

Ans) The term "normal probability distribution" or "normal distribution" refers to a continuous probability distribution for a variable. It is also known as an LAPlace-Gauss distribution or a Gaussian/Gauss distribution. Mean and variance are two factors that affect the normal distribution. The real valued random variables, whose distributions are unknown, are represented by the normal distributions.

In the social and natural sciences, they are employed quite frequently. The normal probability distribution curve, or simply the normal curve, is the graphed representation of the normal distribution. A bell-shaped curve with bilateral symmetry and continuous frequency distribution is known as a normal curve.

Part B

Total Marks: 30

Note: You need to complete the activity as instructed. All the activities are compulsory.

Instructions:

1. Have a title page. Include details like Name, Enrolment number, Email id, Regional Centre,

Study Centre, Programme Title and code, Course title and code and Tutorial code.

2.Use A4 size paper for the tutorial (ruled/ bank).

3. For making tables, blank pages can be used and tables/ graphs (if any) to be drawn in pencil.

4. Content should not be plagiarised.

5. Tutorial should be handwritten.

Activity 1: Collect data from 30 mothers by asking them their present age and the present age of their first-born child. Compute mean, median, mode and standard deviation for each of the groups (mother and children). Compute Pearson Product moment correlation to find out correlationship between the ages of the mothers and their first born. Ensure that you tabulate the raw data and show the calculations with the help of formulae discussed in the course material of BPCC104. 20 Marks

Ans) Consider comparing the average age of students in two schools to see which one has the older students. We cannot draw any conclusions if we compare students on an individual basis. However, if we obtain a representative value for the provided data that denotes the data's characteristics, the comparison is made simple.

Median

We first arrange the given data values of the observations in ascending order. Then, if n is odd, the median is the (n+1/2). And if n is even, then the median will be the average of the n/2 th and the (n/2 +1)th observation.

Formula for Calculating Median:

Median, Me = l + {h x (N/2 – cf )/f}

Where,

l = lower limit of median class.

h=width of median class.

f = frequency of median class,

cf = cumulative frequency of the class preceding the median class.

N = ∑f_i

Mode

That value is the most prevalent one for the variate. In more technical terms, the value at which the data are most concentrated is the variable's mode.

Modal Class: The class with the highest frequency in a frequency distribution is referred to as the modal class.

Formula for Calculating Mode: 

Mo = xk +h{(fk – fk-1)/(2fk -fk-1-fk+1)}

where,

xk = lower limit of the modal class interval.

fk = frequency of the modal class.

fk-1= frequency of the class preceding the modal class.

fk+1 = frequency of the class succeeding the modal class.

h=width of the class interval.

Standard Deviation

The computation of the mean using the aforementioned methods becomes laborious when x and f have large values. In these situations, we employ the step-deviation method described below.

1.  For each class interval, calculate the class mark x, where X = 1/2 (lower limit + upper limit).

2. Choose a suitable value of x, in the middle of the x, column as the assumed mean and denote it by A.

3. Calculate h = [(upper limit)-(lower limit)], which is the same for all the classes.

4. Calculate ui = (xi -A) /h for each class.

5. Calculate fu for each class and hence find ∑(fi x ui).

6. Calculate the mean by using the formula: x = A + {h x ∑(fi x ui)/ ∑fi}

Activity 2: Take block 2 of BPCC104 and write a review of the same in about 1000 words. The review needs to be in your own words and should cover the important aspects of the unit. Also include in review the relevance of measures of central tendency and measures of variability in research. 10 Marks

Ans) Measures of Central Tendency

Mean / Average – sum of all data points or observations in a dataset divided by the total number of data points or observations in the dataset.

The mean or average of this dataset with 5 numbers {2, 4, 6, 8, 10} is: 6

Sum of all data points:                                                         (2+4+6+8+10)

Divided by:                                                                         ———————–  = 6

Number of data points:                                                        5

The median is the midpoint of the values when the values in the dataset are listed in increasing ascending order; there are an equal number of data points above and below the median. The median value will be a single midpoint value if the dataset has an odd number of data points. The median value is the mean/average of the two midpoint values when the number of data points in the dataset is even.

The median of the same dataset {2, 4, 6, 8, 10} is:  6

This dataset has an odd number of data points (5), and the middle data point is the value 6, with 2 numbers below (2, 4) and 2 numbers above (8, 10).

Using an example of a dataset with an even number of data points: The median of this dataset {2, 4, 6, 8, 10, 12} is: (6 + 8) / 2 = 7

Since there are 2 middle data points (6, 8), then we need to calculate the mean of those 2 numbers to determine the median.

Mode – the data point that appears the most times in the dataset.

Using our original dataset {2, 4, 6, 8, 10}, since each of the values only appear once, none appearing more times than the others, this dataset does not have a mode.

Using a new dataset {2, 2, 4, 4, 4, 4, 6, 8, 8, 8, 10}, the Mode in this case is: 4 4 is the value that appears the most times in the dataset.