Cronbach’s Alpha Reliability Analysis of Measurement Scales

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Cronbach’s Alpha Reliability Analysis

Cronbach’s Alpha Reliability analysis is used to study the properties of measurement scales (Likert scale questionnaire) and the items (questions) that make them up. The reliability analysis method computes several commonly used measures of scale reliability. The reliability analysis also provides information about the relationships between individual items in the scale. The intraclass correlation coefficients can be used to compute the interrater reliability estimates.

Consider that you want to know if my questionnaire measures customer satisfaction in a useful way. For this purpose, you can use the reliability analysis to determine the extent to which the items (questions) in your questionnaire are correlated with each other. The overall index of the reliability or internal consistency of the scale as a whole can be obtained. You can also identify problematic items that should be removed (deleted) from the scale.

As an example open the data “satisf.save” already available in SPSS sample files. To check the reliability of Likert scale items follow the steps given below:

Cronbach's Alpha Reliability
Cronbach's Alpha Reliability Analysis Dialog box

Step 1: On the Menu bar of SPSS, Click Analyze > Scale > Reliability Analysis… option

Step 2: Select two more variables that you want to test and shift them from the left pan to the right pan of the reliability analysis dialogue box. Note, that multiple variables (items) can be selected by holding down the CTRL key and clicking the variable you want. Clicking the arrow button between the left and right pan will shift the variables to the item pan (right pan).

Step 3: Click on the “Statistics” Button to select some other statistics such as descriptives (for item, scale, and scale if item deleted), summaries (for means, variances, covariances, and correlations), inter-item (for correlations and covariances) and ANOVA table (for none, F-test, Friedman chi-square and Cochran chi-square) statistics etc.

Reliability Statistics

Click on the “Continue” button to save the current statistics options for analysis. Click the OK button in the Reliability Analysis dialogue box to get the analysis to be done on selected items. The output will be shown in SPSS output windows.

Reliability Analysis Output

The Cronbach’s Alpha Reliability ($\alpha$) is about 0.827, which is good enough. Note that, deleting the item “organization satisfaction” will increase the reliability of remaining items to 0.860.

A rule of thumb for interpreting alpha for dichotomous items (questions with two possible answers only) or Likert scale items (questions with 3, 5, 7, or 9, etc items) is:

  • If Cronbach’s Alpha is $\ge 0.9$, the internal consistency of scale is Excellent.
  • If Cronbach’s Alpha is $0.90 > \alpha \ge 0.8$, the internal consistency of scale is Good.
  • If Cronbach’s Alpha is $0.80 > \alpha \ge 0.7$, the internal consistency of scale is Acceptable.
  • If Cronbach’s Alpha is $0.70 > \alpha \ge 0.6$, the internal consistency of scale is Questionable.
  • If Cronbach’s Alpha is $0.60 > \alpha \ge 0.5$, the internal consistency of scale is Poor.
  • If Cronbach’s Alpha is $0.50 > \alpha $, the internal consistency of scale is Unacceptable.

However, the rules of thumb listed above should be used with caution since Cronbach’s Alpha reliability is sensitive to the number of items in a scale. A larger number of questions can result in a larger Alpha Reliability, while a smaller number of items may result in smaller $\alpha$.

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Standard Deviation: A Measure of Dispersion (2017)

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The standard deviation is a widely used concept in statistics and it tells how much variation (measure of spread or dispersion) is in the data set. It can be defined as the positive square root of the mean (average) of the squared deviations of the values from their mean.
To calculate the standard deviation one has to follow these steps:

Calculation of Standard Deviation

  1. First, find the mean of the data.
  2. Take the difference of each data point from the mean of the given data set (which is computed in step 1). Note that, the sum of these differences must be equal to zero or near to zero due to rounding of numbers.
  3. Now compute the square of the differences obtained in Step 2, it would be greater than zero, and it will be a positive quantity.
  4. Now add up all the squared quantities obtained in step 3. We call it the sum of squares of differences.
  5. Divide this sum of squares of differences (obtained in step 4) by the total number of observations (available in data) if we have to calculate population standard deviation ($\sigma$). If you want t to compute sample standard deviation ($S$) then divide the sum of squares of differences (obtained in step 4) by the total number of observations minus one ($n-1$) i.e. the degree of freedom. Note that $n$ is the number of observations available in the data set.
  6. Find the square root (also known as under root) of the quantity obtained in step 5. The resultant quantity in this way is known as the standard deviation (SD) for the given data set.

The sample SD of a set of $n$ observation, $X_1, X_2, \cdots, X_n$ denoted by $S$ is

\begin{aligned}
\sigma &=\sqrt{\frac{\sum_{i=1}^n (X_i-\overline{X})^2}{n}}; Population\, SD\\
S&=\sqrt{ \frac{\sum_{i=1}^n (X_i-\overline{X})^2}{n-1}}; Sample\, SD
\end{aligned}

The standard deviation can be computed from variance too.

The real meaning of the standard deviation is that for a given data set 68% of the data values will lie within the range $\overline{X} \pm \sigma$ i.e. within one standard deviation from the mean or simply within one $\sigma$. Similarly, 95% of the data values will lie within the range $\overline{X} \pm 2 \sigma$ and 99% within $\overline{X} \pm 3 \sigma$.

Standard Deviation

Examples

A large value of SD indicates more spread in the data set which can be interpreted as the inconsistent behaviour of the data collected. It means that the data points tend to be away from the mean value. For the case of smaller standard deviation, data points tend to be close (very close) to the mean indicating the consistent behavior of the data set.

The standard deviation and variance are used to measure the risk of a particular investment in finance. The mean of 15% and standard deviation of 2% indicates that it is expected to earn a 15% return on investment and we have a 68% chance that the return will be between 13% and 17%. Similarly, there is a 95% chance that the return on the investment will yield an 11% to 19% return.

measures-of-dispersion

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Classical Probability: Example, Definition, and Uses (2017)

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Classical probability is the statistical concept that measures the likelihood (probability) of something happening. In a classic sense, it means that every statistical experiment will contain elements that are equally likely to happen (equal chances of occurrence of something). Therefore, the concept of classical probability is the simplest form of probability that has equal odds of something happening.

Classical Probability Examples

Example 1: The typical example of classical probability would be rolling a fair die because it is equally probable that the top face of the die will be any of the 6 numbers on the die: 1, 2, 3, 4, 5, or 6.

Example 2: Another example of classical probability would be tossing an unbiased coin. There is an equal probability that your toss will yield either head or tail.

Example 3: In selecting bingo balls, each numbered ball has an equal chance of being chosen.

Example 4: Guessing a multiple choice quiz (MCQs) test with (say) four possible answers A, B, C, or D. Each option (choice) has the same odds (equal chances) of being picked (assuming you pick randomly and do not follow any pattern).

Classical Probability Formula

The probability of a simple event happening is the number of times the event can happen, divided by the number of possible events (outcomes).

Mathematically $P(A) = \frac{f}{N}$,

where, $P(A)$ means “probability of event A” (event $A$ is whatever event you are looking for, like winning the lottery, that is event of interest), $f$ is the frequency, or number of possible times the event could happen and $N$ is the number of times the event could happen.

For example,  the odds of rolling a 2 on a fair die are one out of 6, (1/6). In other words, one possible outcome (there is only one way to roll a 1 on a fair die) is divided by the number of possible outcomes.

Classical probability can be used for very basic events, like rolling a dice and tossing a coin, it can also be used when the occurrence of all events is equally likely. Choosing a card from a standard deck of cards gives you a 1/52 chance of getting a particular card, no matter what card you choose. On the other hand, figuring out whether will it rain tomorrow or not isn’t something you can figure out with this basic type of probability. There might be a 15% chance of rain (and therefore, an 85% chance of it not raining).

Classical Probability Formula

Other Examples of classical Probability

There are many other examples of classical probability problems besides rolling dice. These examples include flipping coins, drawing cards from a deck, guessing on a multiple-choice test, selecting jellybeans from a bag, choosing people for a committee, etc.

Classical Probability cannot be used:

Dividing the number of events by the number of possible events is very simplistic, and it isn’t suited to finding probabilities for a lot of situations. For example, natural events like weights, heights, and test scores need normal distribution probability charts to calculate probabilities. Most “real life” things aren’t simple events like coins, cards, or dice. You’ll need something more complicated than classical probability theory to solve them.

It is important to note that the classical probability is most applicable in situations where:

  • All possible outcomes can be clearly defined and listed.
  • Each outcome has an equal chance of happening.

In conclusion, classical probability provides a foundational understanding of probability concepts, and it has various applications in games of chance, simple random sampling, and other situations where clear, equally likely outcomes can be defined.

For further Details see Introduction to Probability Theory

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Skewness in Statistics A Measure of Asymmetry (2017)

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The article is about Skewness in Statistics, a measure of asymmetry. Skewed and skew are widely used terminologies that refer to something that is out of order or distorted on one side. Similarly, when referring to the shape of frequency distributions or probability distributions, the term skewness also refers to the asymmetry of that distribution. A distribution with an asymmetric tail extending out to the right is referred to as “positively skewed” or “skewed to the right”. In contrast, a distribution with an asymmetric tail extending out to the left is “negatively skewed” or “skewed to the left”.

Skewness in Statistics A measure of Asymmetry

Skewness in Statistics

It ranges from minus infinity ($-\infty$) to positive infinity ($+\infty$). In simple words, skewness (asymmetry) is a measure of symmetry, or in other words, skewness is a lack of symmetry.

Skewness by Karl Pearson

Karl Pearson (1857-1936) first suggested measuring skewness by standardizing the difference between the mean and the mode, such that, $\frac{\mu-mode}{\text{standard deviation}}$. Since population modes are not well estimated from sample modes, therefore Stuart and Ord, 1994 suggested that one can estimate the difference between the mean and the mode as being three times the difference between the mean and the median. Therefore, the estimate of skewness will be $$\frac{3(M-median)}{\text{standard deviation}}$$. Many of the statisticians use this measure but after eliminating the ‘3’, that is, $$\frac{M-Median}{\text{standard deviation}}$$. This statistic ranges from $-1$ to $+1$. According to Hildebrand, 1986, absolute values above 0.2 indicate great skewness.

Fisher’s Skewness

Skewness has also been defined concerning the third moment about the mean, that is $\gamma_1=\frac{\sum(X-\mu)^3}{n\sigma^3}$, which is simply the expected value of the distribution of cubed $Z$ scores, measured in this way is also sometimes referred to as “Fisher’s skewness”. When the deviations from the mean are greater in one direction than in the other direction, this statistic will deviate from zero in the direction of the larger deviations.

From sample data, Fisher’s skewness is most often estimated by: $$g_1=\frac{n\sum z^3}{(n-1)(n-2)}$$. For large sample sizes ($n > 150$), $g_1$ may be distributed approximately normally, with a standard error of approximately $\sqrt{\frac{6}{n}}$. While one could use this sampling distribution to construct confidence intervals for or tests of hypotheses about $\gamma_1$, there is rarely any value in doing so.

Bowleys’ Coefficient of Skewness

Arthur Lyon Bowley (1869-19570, has also proposed a measure of asymmetry based on the median and the two quartiles. In a symmetrical distribution, the two quartiles are equidistant from the median but in an asymmetrical distribution, this will not be the case. The Bowley’s coefficient of skewness is $$\frac{q_1+q_3-2\text{median}}{Q_3-Q_1}$$. Its value lies between 0 and $\pm1$.

The most commonly used measures of Asymmetry (those discussed here) may produce some surprising results, such as a negative value when the shape of the distribution appears skewed to the right.

Impact of Lack of Symmetry

Researchers from the behavioral and business sciences need to measure the lack of symmetry when it appears in their data. A great amount of asymmetry may motivate the researcher to investigate the existence of outliers. When making decisions about which measure of the location to report and which inferential statistic to employ, one should take into consideration the estimated skewness of the population. Normal distributions have zero skewness. Of course, a distribution can be perfectly symmetric but may be far away from the normal distribution. Transformations of variables under study are commonly employed to reduce (positive) asymmetry. These transformations may include square root, log, and reciprocal of a variable.

In summary, by understanding and recognizing how skewness affects the data, one can choose appropriate analysis methods, gain more insights from the data, and make better decisions based on the findings.

FAQs About Skewness

  1. What statistical measure is used to find the asymmetry in the data?
  2. Define the term Skewness.
  3. What is the difference between symmetry and asymmetry concept?
  4. Describe negative and positive skewness.
  5. What is the difference between left-skewed and right-skewed data?
  6. What is a lack of symmetry?
  7. Discuss the measure proposed by Karl Pearson.
  8. Discuss the measure proposed by Bowley’s Coefficient of Skewness.
  9. For what distribution, the skewness is zero?
  10. What is the impact of transforming a variable?

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