Estimation and Types of Estimation in Statistics

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The Post is about Introduction to Estimation and Types of Estimation in the Subject of Statistics. Let us discuss Estimation and Types of Estimation in Statistics.

The procedure of making a judgment or decision about a population parameter is referred to as statistical estimation or simply estimation.  Statistical estimation procedures provide estimates of population parameters with a desired degree of confidence. The degree of confidence can be controlled in part, by the size of the sample (larger sample greater accuracy of the estimate) and by the type of estimate made. Population parameters are estimated from sample data because it is not possible (it is impracticable) to examine the entire population to make such an exact determination.

The Types of Estimation in Statistics for the estimation of the population parameter are further divided into two groups (i) Point Estimation and (ii) Interval Estimation

Point Estimation

The objective of point estimation is to obtain a single number from the sample which will represent the unknown value of the population parameter. Population parameters (population mean, variance, etc) are estimated from the corresponding sample statistics (sample mean, variance, etc).
A statistic used to estimate a parameter is called a point estimator or simply an estimator, the actual numerical value obtained by an estimator is called an estimate.

A population parameter is denoted by $\theta$ which is an unknown constant. The available information is in the form of a random sample $x_1,x_2,\cdots,x_n$ of size $n$ drawn from the population. We formulate a function of the sample observation $x_1,x_2,\cdots,x_n$. The estimator of $\theta$ is denoted by $\hat{\theta}$. The different random sample provides different values of the statistics $\hat{\theta}$. Thus $\hat{\theta}$ is a random variable with its sampling probability distribution.

Interval Estimation

A point estimator (such as sample mean) calculated from the sample data provides a single number as an estimate of the population parameter, which can not be expected to be exactly equal to the population parameter because the mean of a sample taken from a population may assume different values for different samples. Therefore, we estimate an interval/ range of values (set of values) within which the population parameter is expected to lie with a certain degree of confidence. This range of values used to estimate a population parameter is known as interval estimate or estimate by a confidence interval, and is defined by two numbers, between which a population parameter is expected to lie.

For example, $a<\bar{x}<b$ is an interval estimate of the population mean $\mu$, indicating that the population mean is greater than $a$ but less than $b$. The purpose of an interval estimate is to provide information about how close the point estimate is to the true parameter.

Types of Estimation

Note that the information developed about the shape of a sampling distribution of the sample mean i.e. Sampling Distribution of $\bar{x}$ allows us to locate an interval that has some specified probability of containing the population mean $\mu$.

Interval Estimate formula when $n>30$ and Population is normal $$\bar{x} \pm Z \frac{\sigma}{\sqrt{n}}$$

Interval Estimate formula when $n<30$ and Population is not normal $$\bar{x} \pm t_{(n-1, \alpha)}\,\, \frac{s}{\sqrt{n}}$$

Which of the two types of estimation in Statistics, do you like the most, and why?

The Types of Estimation in Statistics are as follows:

  • Point estimation is nice because it provides an exact point estimate of the population value. It provides you with the single best guess of the value of the population parameter.
  • Interval estimation is nice because it allows you to make statements of confidence that an interval will include the true population value.

Read about the Advantages of Interval Estimation in Statistics

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Rules for Skewed Data Free Guide

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Introduction to Skewed Data: Lack of Symmetry

Skewness is the lack of symmetry (lack of normality) in a probability distribution. The skewness is usually quantified by the index as given below

$$s = \frac{\mu_3}{\mu_2^{3/2}}$$

where $\mu_2$ and $\mu_3$ are the second and third moments about the mean.

This index formula described above takes the value zero for a symmetrical distribution. A distribution is positively skewed when it has a longer and thin tail to the right. A distribution is negatively skewed when it has a longer thin tail to the left.

Any distribution is said to be skewed when the data points cluster more toward one side of the scale than the other. Creating such a curve that is not symmetrical.

Skewed Data

Skewed Data

The two general rules for Skewed Data are

  1. If the mean is less than the median, the data are skewed to the left, and
  2. If the mean is greater than the median, the data are skewed to the right.

Therefore, if the mean is much greater than the median the data are probably skewed to the right.

Misinterpretation of Mean and Median: The mean can be sensitive to outliers in skewed distributions and might not accurately represent the “typical” value. The median, which is the middle value when the data is ordered, can be a more robust measure of the central tendency for skewed data.

Statistical Tests: Some statistical tests assume normality (zero skewness). If the data is skewed, alternative tests or transformations might be necessary for reliable results.

Identifying Skewed Data

There are a couple of ways to identify skewed data:

  • Visual Inspection: Histograms and box plots are useful tools for visualizing the distribution of the data. Skewed distributions will show an asymmetry in the plots.
  • Skewness Coefficient: This statistic measures the direction and magnitude of the skew in the distribution. A positive value indicates a positive skew, a negative value indicates a negative skew, and zero indicates a symmetrical distribution.
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Interval Estimation and Point Estimation: A Quick Guide 2012

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The problem with using a point estimate is that although it is the single best guess you can make about the value of a population parameter, it is also usually wrong. Interval estimate overcomes this problem using interval estimation technique which is based on point estimate and margin of error.

Interval Estimation

Point Estimation

Point estimation involves calculating a single value from sample data to estimate a population parameter. The examples of point estimation are: (i) Estimating the population mean using the sample mean and (ii) Estimating the population proportion using the sample proportion. The common point estimators are:

  • Sample mean $\overline{x}$ for population mean ($\mu$).
  • Sample proportion ($\hat{p}$​) for population proportion ($P$).
  • Sample variance ($s^2$) for population variance ($\sigma^2$).

Interval Estimation

Interval estimation involves calculating a range of values (set of values: an interval) from sample data to estimate a population parameter. The range constructed has a specified level of confidence. The Components of an interval are:

  • Confidence level: The probability that the true population parameter lies within the interval.
  • Margin of error: The maximum allowable error (difference between the point estimate and the true population parameter).

The common confidence intervals for the population mean are:

  • Confidence interval for a large sample (or known population standard deviation):
    $\overline{x} \pm Z_{\alpha/2} \frac{s}{\sqrt{n}}$
  • Confidence interval for small sample (or unknown population standard deviation):
    $\overline{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$
  • Confidence interval for the population proportion
    $\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p} {1-\hat{p}}}{n}}$

Advantages of Interval Estimation

  • A major advantage of using interval estimation is that you provide a range of values with a known probability of capturing the population parameter (e.g. if you obtain from SPSS a 95% confidence interval you can claim to have 95% confidence that it will include the true population parameter.
  • An interval estimate (i.e., confidence intervals) also helps one not to be so confident that the population value is exactly equal to the single-point estimate. That is, it makes us more careful in interpreting our data and helps keep us in proper perspective.
  • Perhaps the best thing to do is to provide both the point estimate and the interval estimate. For example, our best estimate of the population mean is the value of $32,640 (the point estimate) and our 95% confidence interval is $30,913.71 to $34,366.29.
  • By the way, note that the bigger your sample size, the more narrow the confidence interval will be.
  • Remember to include many participants in your research study if you want narrow (i.e., exact) confidence intervals.

In essence, interval estimation is a game-changer in the field of statistics. Interval estimation, acknowledges the uncertainty inherent in data, providing a range of probable values (interval estimates) instead of a single (point estimate), potentially misleading, point estimate. By incorporating it into the statistical analysis, one can gain a more realistic understanding of the data and can make more informed decisions based on evidence, not just a single number.

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Scatter Diagram: Graphical Representation (2012)

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A scatterplot (also called a scatter graph or scatter Diagram) is used to observe the strength and direction between two quantitative variables. In statistics, the quantitative variables follow the interval or ratio scale from measurement scales.

Scatter Diagram

Usually, in a scatter, diagram the independent variable (also called the explanatory, regressor, or predictor variable) is taken on the X-axis (the horizontal axis) while on the Y-axis (the vertical axis) the dependent (also called the outcome variable) is taken to measure the strength and direction of the relationship between the variables. However, it is not necessary to take explanatory variables on the X-axis and outcome variables on the Y-axis. Because, the scatter diagram and Pearson’s correlation measure the mutual correlation (interdependencies) between the variables, not the dependence or cause and effect.

The diagram below describes some possible relationships between two quantitative variables ($X$ & $Y$). A short description is also given of each possible relationship.

Scatter diagram

A scatter diagram can be drawn between two quantitative variables. The length (number of observations) of both of the variables should be equal. Suppose, we have two quantitative variables $X$ and $Y$. We want to observe the strength and direction of the relationship between these two variables. It can be done in R language easily.

x <- c(5, 7, 8, 7, 2, 2, 9, 4, 11 ,12, 9, 6)
y <- c(99, 86, 87, 88, 111, 103, 87, 94, 78, 77, 85, 86)

plot(x, y)
Scatter Diagram

From the above discussion, it is clear that the main objective of a scatter diagram is to visualize the linear or some other type of relationship between two quantitative variables. The visualization may also help to depict the trends, strength, and direction of the relationship between variables.

Limitations of Scatter Diagrams

  • Limited to Two Variables: Scatter plots can only depict the relationship between two variables at a time. If there are more than two variables, one might need to use other visualization techniques.
  • Strength of Correlation: While scatter diagrams can show the direction of a relationship, they don’t necessarily indicate the strength of that correlation. You might need to calculate correlation coefficients to quantify the strength.

In conclusion, scatter diagrams are a powerful and versatile tool for exploring relationships between variables. By understanding how to create and interpret them, one can gain valuable insights from the data and inform decision-making processes across various disciplines.

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