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

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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

Drawing Scatter Plot/ 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.

Importance of Scatter Diagram

1. Identifies Relationships Between Variables

  • This diagram shows whether two quantitative variables are positively correlated (both increase together), negatively correlated (one increases while the other decreases), or not correlated.
  • It helps in detecting non-linear relationships (e.g., quadratic or exponential trends).

2. Detects Outliers

  • It is used to reveal unusual data points (observations) that deviate from the general trend, which may indicate errors or special cases.

3. Useful for Predictive Analysis

  • It helps in regression analysis by determining if a linear or other model fits the data well.

4. Visualizes Data Distribution

  • It is used to show the spread/scatteredness and clustering of data points, helping in understanding variability.

5. Supports Decision-Making

  • It is used in business, science, engineering, and healthcare to analyze cause-and-effect relationships (e.g., marketing spend vs. sales, temperature vs. product defects).

6. Easy to Interpret

  • Scatter diagrams provide a simple, intuitive way to observe trends without complex statistics.

Example and Uses

  • Business: Analyzing sales vs. advertising expenditure.
  • Quality Control: Checking if machine speed affects defect rates.
  • Healthcare: Studying the relationship between exercise and blood pressure.
  • Psychology: studying the relationship between anxiety score and depression score.

FAQs About Correlation Analysis

  • What is the coefficient of correlation?
  • What is the use of a scatter diagram?
  • What are the limitations of a scatter diagram?
  • How scatter diagram can be used to assess the relation and direction of the relationship between variables?
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Pearson’s Correlation Coefficient SPSS

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The post is about performing Pearson’s Correlation Coefficient SPSS. It involves the steps (i) Enter Data (ii) Navigate to Analysis (iii) Select Variables (iv) Choose Pearson (v) Selection options (vi) Run Analysis (vii) Interpret the results

Introduction to Pearson’s Correlation Coefficient

Pearson’s correlation coefficient (or correlation or simply correlation) is used to find the degree of linear relationship between two continuous variables. The value for a correlation coefficient lies between 0.00 (no correlation) and 1.00 (perfect correlation). Generally, correlations above 0.80 are considered pretty high.

Remember:

  1. Correlation is the interdependence of continuous variables, it does not refer to cause and effect.
  2. Correlation is used to determine the linear relationship between variables.
  3. Draw a scatter plot before performing/calculating the correlation (to check the assumptions of linearity)

How to Perform Pearson’s Correlation Coefficient SPSS

The command for correlation is found at Analyze –> Correlate –> Bivariate i.e.

Correlation Coefficient SPSS

The Bivariate Correlation Coefficient SPSS dialog box will be there:

Pearson's Correlation Coefficient SPSS

Select one of the variables that you want to correlate in the left-hand pane of the Bivariate Correlations dialog box and shift it into the Variables pane on the right-hand pan by clicking the arrow button. Now click on the other variable that you want to correlate in the left-hand pane and move it into the Variables pane by clicking on the arrow button

Pearson's Correlation Coefficient SPSS

Correlation Coefficient SPSS Output

Pearson's Correlation Coefficient SPSS

The Correlations table in the output gives the values of the specified correlation tests, such as Pearson’s correlation. Each row of the table corresponds to one of the variables similarly each column also corresponds to one of the variables.

Interpreting Correlation Coefficient

For example, the cell at the bottom row of the right column represents the correlation of depression with depression which is equal to 1.0. Likewise, the cell at the middle row of the middle column represents the correlation of anxiety with anxiety having a correlation value This in in both cases shows that anxiety is related to anxiety similarly depression is related to depression, so have the perfect relationship.

The cell in the middle row and right column (or the cell in the bottom row at the middle column) is more interesting. This cell represents the correlation between anxiety and depression (or depression with anxiety). There are three numbers in these cells.

  1. The top number is the correlation coefficient value which is 0.310.
  2. The middle number is the significance of this correlation which is 0.018.
  3. The bottom number, 46 is the number of observations that were used to calculate the correlation coefficient between the variables of the study.

Note that the significance tells us whether we would expect a correlation that was this large purely due to chance factors and not due to an actual relation. In this case, it is improbable that we would get an r (correlation coefficient) this big if there was not a relation between the variables.

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