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# Descriptive Statistics

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Descriptive statistics are a form of statistical analysis that are utilised to provide a summary of a dataset. They can be summaries of samples, variables or results.

There are four main types of descriptive statistics that are discussed in further detail below.

## Descriptive statistics in research?

Descriptive statistics allow researchers to provide a basic summary of datasets. The tests are usually carried out before carrying out statistical analysis that tests the hypothesis. These tests are beneficial as they provide researchers with information about potential relationships between variables and information regarding which statistical tests would be appropriate for testing the proposed hypothesis.

It is important to note that descriptive statistics provide information about the dataset and it is not appropriate to generalise to the general population.

### Measures of frequency

The purpose of frequency statistics is to calculate the occurrence of variables, for example, the number of participants in a trial group vs. a control group, or the number of males versus females in a sample. This data is usually outputted in the form of frequency tables.

An example of a frequency table output could provide frequency statistics for two variables: gender and ethnicity. The table would indicate the number of participants classified in each sub-group of the research (N). The table would also provide statistical information regarding how much N (for each variable), represents the total sample in the form of percentages. An example of how this would be reported in research is “the study sample consisted of 216 females and 259 males (N = 474)”.

### Measures of central tendency

There are many different statistical tests used to measure central tendency. Measures of central tendency give a single value that is an average of the entire dataset, this is beneficial for large datasets. The three most commonly used are: mean, median and mode.

• Mean: adding all the values together and dividing by the total number of values
• Median: placing the dataset values in numerical order and identifying which is the middle number
• Mode: most common value in the dataset

The mean is the most commonly reported form of descriptive analysis, and is usually written as, “The number of participants recruited in the study was 10, with a M age of 22.8”.

### Measures of variability or dispersion

We can analyse measures of variability or dispersion using range, interquartile range, standard deviation and variance.

• Range: the highest value minus the smallest value
• Interquartile range: the difference between the median value calculated in the first half and second half of a dataset
• Standard deviation: the average distance of a data point from the mean
• Variance: also measures the average distance of a data point from the mean but it is calculated differently

An example of how this would be reported is “The number of participants recruited in the study was 10, aged 18-27 (M = 22.8 & SD = 8.12) ”.

When writing psychology reports, the mean and standard deviation are the most commonly reported descriptive statistic.

### Measures of position

Measures of position analysis are used to identify a singular value and its relation to other values within a dataset.

An example of descriptive tests that identify measures of position are quantiles. Quantiles are measured by numerically ordering values in ascending order. Quantiles separate populations/samples into intervals of equal sizes. This is done so that ranking of specific data points can be identified.

For example, percentiles is when data is split into 100ths and data points are observed within the different sections of the percentiles. For instance, if you are trying to identify the data point at 36%, then the values would be placed in ascending order and the value that is representative of 36% of the data would be identified.

The amount that interval quantiles are split into is relative to an appropriate number determined by the number of values within a dataset. This data provides information about the distribution of data, which is important for later statistical analyses. If data is found to be skewed then non-parametric tests may be used for statistical analysis, these concepts are explained further in other articles.

## Descriptive statistics and inferential statistics

The purpose of descriptive statistics is to provide a summary of a dataset. However, it is also important for researchers to identify if the sample used in research is appropriate to generalise to the general population. Therefore, a general requirement of research is to carry out descriptive statistics and inferential statistics.

An example of an inferential statistic is hypothesis testing. This analysis involves forming a null hypothesis (no significant effect will be observed between variables) and using an appropriate statistical test to identify if there is a relationship between the variables.

If this is found to have a significant effect size then the null hypothesis is accepted. This implies that changes in the dependent variable are likely due to chance or other potential confounding variables rather than the independent variable. Therefore, the alternative hypothesis (expect to observe a difference between the variables) can be considered inapplicable and cannot be generalised to the population.

## Descriptive Statistics - Key takeaways

• Descriptive statistics are a form of statistical analysis that are utilised to provide a summary of a dataset. They can be summaries of samples, variables, or results.
• There are four main types of descriptive statistics, which are: measures of frequency, measures of central tendency, measures of variability or dispersion and measures of position.
• The most common reported descriptive statistics is the mean and range.
• Descriptive statistics concerning measures of position provides information concerning the normality of the distribution of the sample. This is needed to identify what type of statistical analysis can be used later, for instance, parametric or non-parametric tests.
• Descriptive statistics can only provide summary information of datasets. This means that researchers also need to use inferential statistics to identify if the results obtained can be generalised to the general population.

The four main type of descriptive statistics are: measures of frequency, measures of central tendency, measures of variability/dispersion and measures of position.

Descriptive statistics are a form of statistical analysis that is utilised to provide a summary of a dataset. These can be summaries of samples, variables or results.

Descriptive data are various forms of statistics that provide a summary of the data from research. For example, the mean is a measure of central tendency that is used to find the average value of variables/ data. Whereas inferential statistics are data that allows the researcher to identify if the sample/procedure used in research is appropriate to generalise to the general population. The output from hypothesis testing is an example of inferential statistics.

In psychology research the most common reported descriptive statistics is the mean and the range. An example of how this would be reported is “The number of participants recruited in the study was 10, aged 18- 27 (M = 22.8 & SD = 8.12) ”.

The purpose of descriptive statistics is to provide a summary of data from research and can highlight any potential relationships/trends between variables. Moreover, some descriptive statistics can be used to help identify what type of analysis should be done later, for instance, parametric versus non-parametric statistical analysis.

## Final Descriptive Statistics Quiz

Question

What are descriptive statistics?

Descriptive statistics are a form of statistical analysis that is utilised to provide a summary of a dataset. These can be summaries of samples, variables or results.

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Question

What are the benefits of measuring descriptive statistics?

These can be beneficial as they provide researchers with information about potential relationships between variables and statistical tests that could be appropriate to test the hypotheses proposed.

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Question

Where can you find data concerning the N of males and females in a sample?

Frequency table

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What statistical information do tests measuring central tendency tell us?

They give a single value that summarises an average representing the entire dataset.

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Question

Here is an example dataset, calculate the mean, median and mode: 2, 7, 5, 3, 9, 12, 3

Mean - 5.86 (2 d.p), Median - 5, Mode - 3

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Question

Which is the most commonly reported central tendency measurement and how is it reported?

Mean (M = x).

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What are the statistics used to measure variability/dispersion?

Range, interquartile range, standard deviation and variance.

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How is the interquartile range calculated?

The interquartile range is calculated by subtracting the difference between the median value in the first half and second half of a dataset.

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Question

A study recruited 10 participants, and the descriptive analysis indicated the mean as 22.8 and the standard deviation as 8.12. How would this correctly be reported in psychology research?

'There were a total of 10 participants recruited for this study (M = 22.8 & SD = 8.12)'.

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Question

What are percentiles?

Percentiles are when data is split into 100ths and data points are observed within the different sections of the percentiles. For instance, if you are trying to identify the data point at 36%, then the values would be placed in ascending order and the value that is representative of 36% of the data would be identified.

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Question

What tests can researchers carry out to identify if parametric tests can be used?

Researchers can identify if parametric tests can be used for statistical analysis if a normally distributed chart is plotted. For instance, if the bell curve is not skewed and if q-q plots show data to be normally distributed.

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Question

What is the purpose of inferential statistics?

The purpose of inferential statistics is to identify if a sample or procedure used is appropriate to generalise to the general population.

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What are the principles of hypothesis testing?

Hypothesis testing requires researchers to formulate a null and alternative hypothesis. The null hypothesis is then tested using an appropriate statistical test and if found to be significant then the null hypothesis can be accepted. This means that the results are likely due to chance or confounding variables rather than the intended independent variable. From these findings, it can be inferred that results observed from research are inappropriate to be generalised to the population.

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Question

What are the three measures of central tendency?

The three measures of central tendency are mean, median, and mode.

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How do you calculate the mean?

Add up all the values in a data set, and then divide by the total number of values. For example, a data set has the values 2, 4, 6, 8, 10. The mean would be (2+4+6+8+10) ÷ 5 = 6.

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What are the advantages of the mean?

• The mean is a powerful statistic used in population parameters. These population parameters we derive from the mean can be used in inferential statistics.

• The mean is the most sensitive and precise of the three measures of central tendency.

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What are the disadvantages of the mean?

• As the mean is so sensitive it can easily be distorted by unrepresentative values (outliers).

• As the mean is very precise, sometimes the values calculated do not make sense. For example, at a school, the mean number of siblings someone has is 2.4.

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What is the median?

The median is the central number in a data set.

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How do you calculate the median if there is an even number in the data set?

The median is between the two central values. For example, if the central values are 6 and 11, the mean of these two numbers is (6+11) ÷ 2 = 8.5.

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What are the advantages of the median?

• The median is unaffected by extreme values unrepresentative of the data set.

• The median is easier to calculate than the mean.

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What are the disadvantages of the median?

• The median does not take into account the exact distances between values.

• The median cannot be used in estimates of population parameters.

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How do you find out the mode?

The mode is the category with the highest frequency count. For example, for a data set of 3, 4, 5, 6, 6, 6, 7, 8, 8, the mode is 6.

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What are the advantages of the mode?

• Can show which category is the most frequently occurring.

• Unaffected by extreme values unrepresentative of the data set.

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What are the disadvantages of the mode?

• The mode does not take into account the exact distances between values.

• The mode cannot be used in estimates of population parameters.

• Not useful for small data sets which have values that occur equally frequently.

• Not useful for categories with grouped data.

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Question

What are measures of dispersion?

The measure of dispersion is the measure of the spread of scores in a data set. It is the extent to which the values vary around the central or average value.

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Why are measures of dispersion important?

If we don’t know the dispersion, a mean value can be misleading. E.g., two datasets have the same mean, but there is a large difference in the datasets' variation of values.

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How do you calculate the range?

The range is the difference between the highest and lowest values of a data set. For example, if the highest value is 50, and the lowest value is 12, the range would be 50-12 = 38.

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What are the advantages of using the range?

• We are able to include extreme values (outliers) when calculating the range.

• It is easy to calculate

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What are the disadvantages of using the range?

• As extreme scores are included, the range could be distorted.

• The range does not tell us much information about the dispersion of values between the top and bottom scores.

It does not give information about whether the values are close to the mean or more spaced out.

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What is the standard deviation a measure of?

The standard deviation is a measure of the mean distance of scores in a data set from the mean.

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What does a large standard deviation indicate?

The scores are widely spread out above and below the mean, therefore the mean is not representative of the data set.

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What does a small standard deviation indicate?

The mean is a good representation of the scores in the data set.

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What are the advantages of using the standard deviation?

• The SD can be used in estimates of population parameters.

• The SD is the most sensitive measure of dispersion as all values in the data set are taken into account.

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Question

What are the disadvantages of standard deviation?

• The SD is distorted by extreme values.

• It is rather complicated to calculate.

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