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Block 1 Unit 2 Descriptive and Inferential Statistics

1) What is descriptive statistics? 

Descriptive statistics is a branch of statistics, which deals with descriptions of obtained data. On the basis of these descriptions a particular group of population is defined for corresponding characteristics. The descriptive statistics include classification, tabulation, diagrammatic and graphical presentation of data, measures of central tendency and variability. These measures enable the researchers to know about the tendency of data or the scores, which further enhance the ease in description of the phenomena.

Example: A class teacher calculates the class average on their final exam. Was 64%.

2) Discuss advantages and disadvantages of descriptive statistics.

The Advantages of Descriptive statistics are given below:

• It is essential for arranging and displaying data.
• It forms the basis of rigorous data analysis.
• It is easier to work with, interpret, and discuss than raw data.
• It helps in examining the tendencies, variability, and normality of a data set.
• It can be rendered both graphically and numerically.
• It forms the basis for more advanced statistical methods.

The disadvantages of descriptive statistics can be listed as given below:

• It can be misused, misinterpreted, and incomplete.
• It can be of limited use when samples and populations are small.
• It offers little information about causes and effects.
• It can be dangerous if not analysed completely.
• There is a risk of distorting the original data or losing important detail.

3) What do you mean by organisation of data? 

The organisation of data is the initial step in descriptive statistics, where raw data is transformed into a meaningful and manageable format for analysis and interpretation.

4) State different methods of organising raw data.

There are four major statistical techniques for organising the data. These are:

i) Classification

ii) Tabulation

iii) Graphical Presentation, and

iv) Diagrammatical Presentation

5) Define measures of dispersion. 

By knowing only the mean, median or mode, it is not possible to have a complete picture of a set of data. Average does not tell us about how the score or measurements are arranged in relation to the center. It is possible that two sets of data with equal mean or median may differ in terms of their variability. Therefore, it is essential to know how far these observations are scattered from each other or from the mean. Measures of these variations are known as the ‘measures of dispersion’. The most commonly used measures of dispersion are range, average deviation, quartile deviation, variance and standard deviation.

6) Why is it that standard deviation is considered as the best measures of variability?

It is based on all observations. It is amenable to further mathematical treatments. Of all measures of dispersion, standard deviation is least affected by fluctuation of sampling.

7) Explain the importance of inferential statistics.

Inferential statistics are essential in research, allowing us to draw conclusions about populations based on data from smaller samples. This is crucial because studying entire populations is often impractical or impossible.

Inferential statistics enable:

• Generalisation: We can extend findings from a sample to a larger population, making research more meaningful and applicable. For example, exit polls use inferential statistics to predict election results based on a subset of voters.
• Hypothesis Testing: Researchers can test hypotheses about relationships between variables by determining the probability of observing the sample data if no relationship exists.
• Estimation: Inferential statistics allow us to estimate population characteristics, such as the mean or correlation coefficient, using sample data.

The Central Limit Theorem is foundational to inferential statistics. It states that the distribution of sample means becomes normal as the sample size increases, even if the population distribution is not normal. This allows researchers to use methods assuming a normal distribution, even when the population distribution is unknown.

In essence, inferential statistics bridge the gap between limited sample data and the broader population, allowing for more impactful research findings and evidence-based decision-making.

8) Describe the important properties of good estimators.

i) Unbiased: An unbiased estimator is one in which, if we were to obtain an infinite number of random samples of a certain size, the mean of the statistic would be equal to the parameter. The sample mean, ( x ) is an unbiased estimate of population mean (μ) because if we look at possible random samples of size N from a population, then mean of the sample would be equal to μ.

ii) Consistent: A consistent estimator is one that as the sample size increased, the probability that estimate has a value close to the parameter also increased. Because it is a consistent estimator, a sample mean based on 20 scores has a greater probability of being closer to (μ) than does a sample mean based upon only 5 scores.

iii) Accuracy: The sample mean is an unbiased and consistent estimator of population mean (μ). But we should not over look the fact that an estimate is just a rough or approximate calculation. It is unlikely in any estimate that ( x ) will be exactly equal to population mean (μ). Whether or not x is a good estimate of (μ) depends upon the representative ness of sample, the sample size, and the variability of scores in the population.

9) Discuss the different types of hypothesis formulated in hypothesis testing.

1. The probability of chance occurrence of the observed results is examined by the null hypothesis (H0). Null hypothesis is a statement of no differences. The other way to state null hypothesis is that the two samples came from the same population. Here, we assume that population is normally distributed and both the groups have equal means and standard deviations.
2. Since the null hypothesis is a testable proposition, there is counter proposition to it known as alternative hypothesis and denoted by H1. In contrast to null hypothesis, the alternative hypothesis (H1) proposes that:
1. the two samples belong to two different populations,
2. their means are estimates of two different parametric means of the respective population, and
3. there is a significant difference between their sample means

10) Discuss the errors involved in hypothesis testing.

There are two types of errors regarding decision to accept or to reject a null hypothesis.

1. Type I Error: When the null hypothesis is true, a decision to reject it is an error and this kind of error is known as type I error in statistics. The probability of making a type I error is denoted as ‘á’ (read as alpha). The null hypothesis is rejected if the probability ‘p’ of its being correct does not exceed the p. The higher the chosen level of p for considering the null hypothesis, the greater is the probability of type I error.
2. Type II Error: When null hypothesis is false, a decision to accept it is known as type II error. The probability of making a type II error is denoted as ‘â’ (read as beta). The lower the chosen level of significance p for rejecting the null hypothesis, the higher is the probability of the type II error. With a lowering of p, the rejection region as well as the probability of the type I error declines and the acceptance region (1-p) widens correspondingly.

The goodness of a statistical test is measured by the probability of making a type I or type II error. For a fixed sample size n, á and â are so related that reduction in one causes increase in the other. Therefore, simultaneous reductions in á and â are not possible. If n is increased, it is possible to decrease both á and â.

11) Explain the various steps involved in hypothesis testing.

Step 1. Set up a null hypothesis suitable to the problem.

Step 2. Define the alternative hypothesis.

Step 3. Calculate the suitable test statistics.

Step 4. Define the degrees of freedom for the test situation.

Step 5. Find the probability level ‘p’ corresponding to the calculated value of the test statistics and its degree of freedom. This can be obtained from the relevant tables.

Step 6. Reject or accept null hypothesis on the basis of tabulated value and calculated value at practical probability level.

12) What is Inferential statistics

Inferential statistics is a branch of statistics that focuses on drawing conclusions about a population based on data from a sample. It involves using sample data to make inferences or generalisations about the larger population from which the sample was drawn.

Example: In a sample of school children, the investigator found an average IQ was 110.

13) What is Exclusive method of classification

The exclusive method is a technique used in the classification of data, particularly when creating frequency distributions. In this method, the upper limit of a class interval is excluded from that class and becomes the lower limit of the next class interval. This ensures that each data point falls into only one class.

14) What is actual method of classification

The actual method of classification, also known as the true class method, is another technique used in frequency distributions. It recognizes that scores are continuous and extend 0.5 units above and below their face value. This method defines class limits using these actual or true boundaries to enhance precision.

Example:

If we are classifying exam scores using the actual method with class intervals of 10, the actual class limits would be:

• 59.5-69.5
• 69.5-79.5
• 79.5-89.5

A student who scores 69 would fall into the 59.5-69.5 interval, as their score is considered to extend to 69.5.

15) What is Frequency distribution?

A much clear picture of the information of score emerges when the raw data are organised as a frequency distribution. Frequency distribution shows the number of cases following within a given class interval or range of scores. A frequency distribution is a table that shows each score as obtained by a group of individuals and how frequently each score occurred.

16) Explain Estimation.

Estimation is a fundamental process in inferential statistics where we use information from a sample to make an educated guess about the value of an unknown population parameter. A parameter is a numerical characteristic of a population, such as the population mean (μ) or the population standard deviation (σ). Since it's usually impractical to measure an entire population, we rely on samples to estimate these parameters.

17) Explain Point Estimation.

Point estimation is a type of estimation where we use a single value, calculated from the sample data, to estimate the unknown population parameter. It's like aiming for a bullseye on a dartboard - you're trying to get as close to the true value as possible with a single shot.

18) Explain Interval Estimation.

Interval estimation is a type of estimation where we construct a range of values, called a confidence interval, that is likely to contain the unknown population parameter. It's like casting a net rather than throwing a single dart – you're increasing your chances of capturing the true value within a range.

19) Explain Histogram.

A histogram is a graphical representation of a frequency distribution of continuous data. It displays the distribution of numerical data by grouping data into "bins" or "classes" and representing the frequency of data points in each bin with bars. The height of each bar corresponds to the frequency of data points within that bin.

20) Explain Bar Diagram.

A bar diagram, also known as a bar chart, is used to represent categorical data. It presents data with rectangular bars where the length or height of each bar is proportional to the value it represents. Unlike histograms, the bars in a bar diagram are typically separated to emphasise the distinct nature of the categories.

21) Explain Frequency Polygon.

A frequency polygon is a line graph that represents a frequency distribution. It is constructed by plotting points at the midpoint of each class interval on the horizontal axis, with the height of each point representing the frequency of that interval. These points are then connected with straight lines, forming a polygon.

22) Explain Pie Diagram.

A pie diagram, also known as a pie chart, is a circular chart used to represent the proportions of different categories in a whole. The circle is divided into slices or sectors, where the size of each slice is proportional to the percentage or fraction it represents.

23) Differentiate between following components of a statistical table that is “Caption” and “Stub head” “Head note” and “Foot note”.

• Caption vs. Stub Head
• Captions are concise, self-explanatory headings for columns in a table. They may include headings and sub-headings, and are typically placed in the middle of the columns.
• Stubs provide brief, self-explanatory headings for rows in a table.
• In essence, captions describe the data presented in columns, while stubs describe the data presented in rows.
• Head Note vs. Foot Note
• Head Notes are placed at the extreme right, below the title, and clarify the units of measurement used in the table.
• Foot Notes appear below the table and provide qualifying statements related to the data that are not covered in the title, captions, or stubs. They might explain specific data points, abbreviations, or sources of information.
• In short, head notes explain the units used in the table, while footnotes offer additional information or context for the data presented.

24) What is Statistical Inference?

Statistical inference is the process of drawing conclusions about a population based on observations made on a sample drawn from that population. It is a powerful tool that allows researchers to generalise findings from a smaller group to a larger group, even when it's not feasible to study the entire population. The sources emphasize that statistical inference is widely applicable in various fields, including psychology.

For example, in psychology, researchers might use statistical inference to:

• Determine if a new therapy is effective in treating anxiety by comparing the outcomes of a treatment group and a control group.
• Estimate the prevalence of depression in a certain population based on a survey of a representative sample.
• Predict election results based on exit polls of a subset of voters.

25) What are the Procedures Involved in Statistical Inference?

The sources describe two main procedures involved in statistical inference:

• Estimation: In estimation, the goal is to make an educated guess about the value of an unknown population parameter based on the information from a sample. There are two types of estimation:
• Point estimation: Using a single value, such as the sample mean, to estimate the population parameter.
• Interval estimation: Constructing a range of values, called a confidence interval, that is likely to contain the population parameter.
• Hypothesis testing: In hypothesis testing, a specific claim or hypothesis about the population is tested using sample data. The goal is to determine whether there is enough evidence from the sample to support or reject the hypothesis.

Important Points

i) Statistics in plural means data.

ii) Statistics in singular means Statistical Methods.

iii) Data collection is first step in statistics.

iv) The last step in statistics is interpretation.

v) Alternative hypothesis is a statement of significant difference.

vi) Null hypothesis is denoted by H0

vii) Alternative hypothesis is not directly tested statistically.

viii) Level of Significance is that probability of chance of occurrence of observed results.

ix) One tail test is a directional statistical test.

x) When the null hypothesis is true, a decision to reject is known as Type 1 Error.

xi) When a null hypothesis is false, a decision to accept is known as Type 2 Error.

xii) Mean is affected by extreme values

xiii) Mode is not affected by extreme values

xiv) Mode is not useful in studying qualitative facts such as intelligence

xv) Median is not affected by extreme values

xvi) Range is an unstable measure of variability

xvii) Standard deviation is most suitable measures of dispersion

xviii) Skewness can be positive, negative or zero.