Statistics & Probability

Statistics & Probability- Drawing valid statistical conclusions

Drawing statistical conclusions involves the usage of enumerative and analytical studies, which are

• Enumerative or descriptive studies – It describes data using math and graphs and focus on the current situation like a tailor taking a measurement of length, is obtaining quantifiable information which is an enumerative approach.
• Analytical (Inferential) Studies – The objective of statistical inference is to draw conclusions about population characteristics based on the information contained in a sample.

Statistical Basic Terms – Various statistics terminologies which are used extensively are

Data – facts, observations, and information that come from investigations.

• Measurement data sometimes called quantitative data — the result of using some instrument to measure something (e.g., test score, weight);
• Categorical data also referred to as frequency or qualitative data. Things are grouped according to some common property(ies) and the number of members of the group are recorded (e.g., males/females, vehicle type).

Variable – property of an object or event that can take on different values. For example, college major is a variable that takes on values like mathematics, computer science, etc.

• Discrete Variable – a variable with a limited number of values (e.g., gender (male/female).
• Continuous Variable – It is a variable that can take on many different values, in theory, any value between the lowest and highest points on the measurement scale.
• Independent Variable – a variable that is manipulated, measured, or selected by the user as an antecedent condition to an observed behavior. In a hypothesized cause-and-effect relationship, the independent variable is the cause and the dependent variable is the effect.
• Dependent Variable – a variable that is not under the user’s control. It is the variable that is observed and measured in response to the independent variable.

Descriptive Statistics
Central Tendencies – Central tendency is a measure that characterizes the central value of a collection of data that tends to cluster somewhere between the high and low values in the data.

• Mean – The mean is the most common measure of central tendency. It is the ratio of the sum of the scores to the number of the scores.
• Median – It divides the distribution into halves; half are above it and half are below it when the data are arranged in numerical order.
• Mode – It is the most frequent or common score in the distribution or the point or value of X that corresponds to the highest point on the distribution.

Measures of Spread – Although the average value in a distribution is informative about how scores are centered in the distribution.

• Range – The simplest measure of variability to compute and understand is the range. The range is the difference between the highest and lowest score in a distribution.
• Inter-quartile Range (IQR) – Provides a measure of the spread of the middle 50% of the scores.
• Variance (σ2) – The variance is a measure based on the deviations of individual scores from the mean.
• Standard deviation (σ) – The standard deviation is defined as the positive square root of the variance and is a measure of variability expressed in the same units as the data.
• Coefficient of variation (cv) – Measures of variability can not be compared like the standard deviation of the production of bolts to the availability of parts.

Measures of Shape – For distributions summarizing data from continuous measurement scales, statistics can be used to describe how the distribution rises and drops.

• Symmetric – Distributions that have the same shape on both sides of the center are called symmetric and those with only one peak are referred to as a normal distribution.
• Skewness – It refers to the degree of asymmetry in a distribution. Asymmetry often reflects extreme scores in a distribution.

Measures of Association – It provides information about the relatedness between variables so as to help estimate the existence of a relationship between variables and it’s strength. They are

• Covariance – It shows how the variable y reacts to a variation of the variable x.
• Correlation coefficient (r) – It is a number that ranges between −1 and +1. The sign of r will be the same as the sign of the covariance.
• Coefficient of determination (r2) – It measures the proportion of changes of the dependent variable y as explained by the independent variable x.

Frequency Distributions – A distribution is the amount of potential variation in the outputs of a process, usually expressed by its shape, mean or variance.

Cumulative Frequency Distribution – It is created from a frequency distribution by adding an additional column to the table called cumulative frequency thus, for each value, the cumulative frequency for that value is the frequency up to and including the frequency for that value.

Central limit theorem and sampling distribution of the mean
The central limit theorem is the basis of many statistical procedures. The theorem states that for sufficiently large sample sizes ( n ≥ 30), regardless of the shape of the population distribution, if samples of size n are randomly drawn from a population that has a mean µ and a standard deviation σ , the samples’ means X are approximately normally distributed.

When means are used as estimators to make inferences about a population’s parameters and n ≥ 30, the estimator will be approximately normally distributed in repeated sampling.

Basic Probability
Basic probability concepts and terminology is discussed below

• Probability – It is the chance that something will occur. It is expressed as a decimal fraction or a percentage. It is the ratio of the chances favoring an event to the total number of chances for and against the event. The probability of getting 4 with a rolling of dice, is 1 (count of 4 in a dice) / 6 = .01667. Probability then can be the number of successes divided by the total number of possible occurrences. Pr(A) is the probability of event A. The probability of any event (E) varies between 0 (no probability) and 1 (perfect probability).
• Sample Space – It is the set of possible outcomes of an experiment or the set of conditions. The sample space is often denoted by the capital letter S. Sample space outcomes are denoted using lower-case letters (a, b, c . . .) or the actual values like for a dice, S={1,2,3,4,5,6}
• Event – An event is a subset of a sample space. It is denoted by a capital letter such as A, B, C, etc. Events have outcomes, which are denoted by lower-case letters (a, b, c . . .) or the actual values if given like in rolling of dice, S={1,2,3,4,5,6}, then for event A if rolled dice shows 5 so, A ={5}. The sum of the probabilities of all possible events (multiple E’s) in total sample space (S) is equal to 1.
• Independent Events – Each event is not affected by any other events for example tossing a coin three times and it comes up “Heads” each time, the chance that the next toss will also be a “Head” is still 1/2 as every toss is independent of earlier one.
• Dependent Events – They are the events which are affected by previous events like drawing 2 Cards from a deck will reduce the population for second card and hence, it’s probability as after taking one card from the deck there are less cards available as the probability of getting a King, for the 1st time is 4 out of 52 but for the 2nd time is 3 out of 51.
• Simple Events – An event that cannot be decomposed is a simple event (E). The set of all sample points for an experiment is called the sample space (S).
• Compound Events – Compound events are formed by a composition of two or more events. The two most important probability theorems are the additive and multiplicative laws.
• Union of events – The union of two events is that event consisting of all outcomes contained in either of the two events. The union is denoted by the symbol U placed between the letters indicating the two events like for event A={1,2} and event B={2,3} i.e. outcome of event A can be either 1 or 2 and of event B is 2 or 3 then, AUB = {1,2}
• Intersection of events – The intersection of two events is that event consisting of all outcomes that the two events have in common. The intersection of two events can also be referred to as the joint occurrence of events. The intersection is denoted by the symbol ∩ placed between the letters indicating the two events like for event A={1,2} and event B={2,3} then, A∩B = {2}
• Complement – The complement of an event is the set of outcomes in the sample space that are not in the event itself. The complement is shown by the symbol ` placed after the letter indicating the event like for event A={1,2} and Sample space S={1,2,3,4,5,6} then A`={3,4,5,6}
• Mutually Exclusive – Mutually exclusive events have no outcomes in common like the intersection of an event and its complement contains no outcomes or it is an empty set, Ø for example if A={1,2} and B={3,4} and A ∩ B= Ø.
• Equally Likely Outcomes – When a sample space consists of N possible outcomes, all equally likely to occur, then the probability of each outcome is 1/N like the sample space of all the possible outcomes in rolling a die is S = {1, 2, 3, 4, 5, 6}, all equally likely, each outcome has a probability of 1/6 of occurring but, the probability of getting a 3, 4, or 6 is 3/6 = 0.5.
• Probabilities for Independent Events or multiplication rule – Independent events occurrence does not depend on other events of sample space then the probability of two events A and B occurring both is P(A ∩ B) = P(A) x P(B) and similarly for many events the independence rule is extended as P(A∩B∩C∩. . .) = P(A) x P(B) x P(C) . . . This rule is also called as the multiplication rule. For example the probability of getting three times 6 in rolling a dice is 1/6 x 1/6 x 1/6 = 0.00463
• Probabilities for Mutually Exclusive Events or Addition Rule – Mutually exclusive events do not occur at the same time or in the same sample space and do not have any outcomes in common. Thus, for two mutually exclusive events, A and B, the event A∩B = Ø, and the probability of events A and B occurring is zero, as P(A∩B) = 0, for events A and B, the probabilities of either or both of the events occurring is P(AUB) = P(A) + P(B) – P(A∩B) also called as addition rule. For example let P(A) = 0.2, P(B) = 0.4, and P(A∩B) = 0.5, then P(AUB) = P(A) + P(B) – P(A∩B) = 0.2 + 0.4 – 0.5 = 0.1
• Conditional probability – It is the result of an event depending on the sample space or another event. The conditional probability of an event (the probability of event A occurring given that event B has already occurred)

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