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Descriptive Statistics

If you were to tell a friend a specific story about something that happened over the weekend, you would probably not start with the most relevant information. Instead, you would begin to describe the setting and the people there and, generally, offer a bit of context. This is what descriptive statistics are used for. Descriptive statistics describe the Raw data and are usually the first step in the data analysis process.

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# Descriptive Statistics

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If you were to tell a friend a specific story about something that happened over the weekend, you would probably not start with the most relevant information. Instead, you would begin to describe the setting and the people there and, generally, offer a bit of context. This is what descriptive statistics are used for. Descriptive statistics describe the Raw data and are usually the first step in the data analysis process.

• We will start by learning the definition of descriptive statistics.
• Moving on from this, we will look at descriptive statistics in psychology, and we will cover some descriptive statistics examples.
• Then, we will look at some descriptive statistical analyses used in psychology.
• Last, you will learn the difference between descriptive and inferential statistics.

## Descriptive Statistics Definition

Descriptive statistics allow researchers to create a preliminary summary of the Raw data. For this reason, they can also be referred to as summary statistics. Descriptive statistics are the first step in data analysis and provide valuable information for choosing the correct statistical test.

Descriptive statistics is a form of statistical analysis utilised to summarise a dataset.

As you can probably deduce from its name, descriptive statistics describe the main aspects of the data. They 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.

Researchers, however, do not conclude from descriptive statistics. To go beyond description and infer results, researchers use inferential statistics.

Descriptive statistics describe raw data, and inferential statistics make predictions about a larger population.

## Descriptive Statistics: Psychology

Descriptive statistics are usually presented graphically, either on tables, frequency distributions, histograms, or bar charts.

Generally, descriptive statistics are used in psychology research to summarise datasets.

However, descriptive statistics cannot be used to make inferences or generalisations about broader populations.

## Descriptive Statistical Analysis: Measures of Frequency Distribution

The frequency distribution describes the number of observations for a possible variable value. This information is often displayed in frequency tables.

Imagine a study looking into the relationship between two Variables: hair colour and nationality. A frequency table would look like this:

 Hair Colour Frequency Nationality Frequency Black 7 Ireland 5 Brown 6 England 15 Blonde 14 Wales 5 Ginger 3 Scotland 5 Total Sample 30 Total Sample 30

From such a table, researchers can state that 14 individuals within the sample were blonde and that 5 were Irish.

## Descriptive Statistical Analysis: 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

Let's consider an example to understand central tendency. Imagine a study looking into the relationship between exam performance and revision time.

The raw data of 10 participants may look like this:

 Participant Number Time Revised (in hours) 1 6 2 3 3 7 4 5 5 6 6 9 7 5 8 6 9 6 10 4

The mean (M) is the number one gets by adding all values together and dividing them by the total number. In this example, the mean amount of hours the sample studied is:

(6 +3 + 7 + 5 + 6 + 9 + 5 + 6 + 6 + 4) / 10 = 5.7 hours.

When reporting more than one means, it is written the following way: The average score of revision time among medical students was higher (M = 8.7) than in philosophy students (M = 5.6).

The numbers need to be placed in sequential numerical order to find the median, and the median is the middle number. In this case, it is:

3, 4, 5, 5, 6, 6, 6, 6, 7, 9. In this example, the median is 6

The mode refers to the most popular score in the data, which in this example is six because it reflects the data of 4 participants.

## Descriptive Statistics Examples: Measures of Variability or Dispersion

Measures of variability are meant to describe the amounts of differences within the data set. It's somehow the opposite of the central tendency.

There are four types of variability measures:

• Range: the highest value minus the smallest value
• Interquartile range: the difference between the median value calculated in the second half and first half of a dataset
• Standard deviation (sd): 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

Let's consider the example above.

 Participant Number (N) Time Revised (in hours) 1 6 2 3 3 7 4 5 5 6 6 9 7 5 8 6 9 6 10 4

The range would be the highest score, 9, minus the lowest score, 3. Therefore, the range in this example is 9 - 3 = 6.

The interquartile range is the difference between the median values calculated in a dataset's first half and second half. The first half of the dataset would be 3, 4, 5, 5, and 6, while the second would be 6, 6, 6, 7, and 9. The median of the first half is 5, and the median of the second half is 6. Therefore the interquartile range is 6 - 5 = 1.

The standard deviation and the variance are slightly more complex to calculate; they measure the distance of a given data point from the mean.

A small variance or standard deviation suggests that the scores do not vary too much from the mean. On the contrary, a high variance or standard deviation indicates that the data is widely spread from the mean.

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

## Descriptive Statistics Examples: Measures of Position

A measure of position identifies the position of a given value from the other values. Quartiles and percentiles are used to measure position.

Percentiles, for example, divide the data into four categories: the 25th, the 50th and the 75th percentile. When calculating percentiles, values need to be put in ascending order. In this way, researchers can establish which scores are associated with the different percentiles.

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.

This data provides information about the distribution of data, which is crucial for later statistical analyses. If data is skewed, Non-Parametric Tests may be used for statistical analysis.

## Descriptive and Inferential Statistics

As you learned, descriptive statistics offer information about a specific dataset. And while these are helpful, psychologists also need other statistical tests to draw conclusions. For this, psychologists use inferential statistics. These are based on probabilities and let researchers test hypotheses and draw conclusions about populations.

Let's consider studying ice cream consumption rates across the year. Descriptive statistics may suggest that more ice cream is consumed in July than in January. And although it may be tempting to conclude that ice cream consumption is lower in January compared to July, this would not be accurate.

In order to make such a statement, one would need to test whether there is a significant difference between the means of ice cream consumption in both months. And this can only be achieved through inferential statistics.

## Descriptive Statistics - Key takeaways

• Descriptive statistics is a form of statistical analysis utilised to summarise a dataset.
• There are four main types of descriptive statistics: measures of frequency, central tendency, variability or dispersion, and measures of position.
• The most commonly reported descriptive statistics are the mean and range.
• Descriptive and inferential statistics have different uses, e.g. the first is used to summarise data and the latter is used to make inferences.

## Frequently Asked Questions about Descriptive Statistics

The four primary types of descriptive statistics are measures of frequency, measures of central tendency, measures of variability/dispersion and measures of position.

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

Descriptive data are various forms of statistics that summarise the data from research. For example, the mean is a central tendency measure used to find the average value of variables/ data. In contrast, inferential statistics are data that allow 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.

The most commonly reported descriptive statistics in psychology research are 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 help identify what type of analysis should be done later, for instance, parametric versus non-parametric statistical analysis.

## Descriptive Statistics Quiz - Teste dein Wissen

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.

Show question

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.

Show question

Question

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

Frequency table

Show question

Question

What statistical information do tests measuring central tendency tell us?

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

Show question

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

Show question

Question

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

Mean (M = x).

Show question

Question

What are the statistics used to measure variability/dispersion?

Range, interquartile range, standard deviation and variance.

Show question

Question

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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Question

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.

Show question

Question

What are the three measures of central tendency?

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

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Question

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, and 10. The mean would be (2+4+6+8+10) ÷ 5 = 6.

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Question

What are the advantages of the mean?

The mean is a powerful statistic used in population parameters. And the mean is the most sensitive and precise of the three measures of central tendency.

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Question

What are the disadvantages of the mean?

The disadvantage of the mean is that as the mean is so sensitive, it can easily be distorted by unrepresentative values (outliers).

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Question

What is the median?

The median is the central number in a ranked data set.

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Question

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.

Show question

Question

What are the advantages of the median?

The advantages of the median are that it is unaffected by extreme outliers and is easier to calculate than, say, the mean.

Show question

Question

What are the disadvantages of the median?

However, the disadvantage of the measure of central tendency is that it doesn’t account for the exact distances between values like the mean does. Furthermore, it can’t be used to make estimations concerning population parameters.

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Question

How do you find 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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Question

What are the advantages of the mode?

The mode’s advantages are that it can show and highlight which category has the most occurrences in a category. Similar to the median, it is unaffected by extreme outliers.

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Question

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. E.g., 5, 6, 7, 8.
• Not useful for categories with grouped data, e.g., 1-4, 5-7, 8-10.

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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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Question

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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Question

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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Question

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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Question

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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Question

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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Question

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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Question

What does a small standard deviation indicate?

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

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Question

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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Question

The      is the most commonly reported descriptive statistic.

mean

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Question

A mean score of 22.8 was found in females and 34 in males. How would this be reported?

On average males received higher scores (M = 34) than females (M = 22.8).

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Question

Descriptive statistics provide             information.

summarative

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Question

Descriptive statistics are carried out whilst conducting statistical tests, true or false?

False

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Question

Frequency tables show how       the variable occurs.

often

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Question

What symbol is used to show the number of participants in research?

N

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Question

The range is an example of a measure of tendency, true or false?

False

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Standard deviation is often reported alongside the mean, true or false?

True

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Question

To calculate the standard deviation, do we need to know the mean?

Yes

Show question

Question

When calculating percentiles, values need to be put in            order.

ascending

Show question

Question

Percentiles are split into     ths.

100

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Question

Calculate the mean of the following dataset: 2, 4, 6, 6, 6, 12, 8, 9, 1.

6.

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Question

What is the mode in the following dataset: 2, 4, 6, 6, 6, 12, 8, 9, 1?

6.

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What is the median in the following dataset: 2, 4, 6, 6, 6, 12, 8, 9, 1?

6.

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Question

What is the median in the following dataset: 2, 4, 6, 6, 6, 12, 8, 9?

6.5.

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Question

Population parameters derived from the mean can be used in            statistics.

inferential.

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## Test your knowledge with multiple choice flashcards

The      is the most commonly reported descriptive statistic.

Descriptive statistics are carried out whilst conducting statistical tests, true or false?

The range is an example of a measure of tendency, true or false?

Flashcards in Descriptive Statistics74

Start learning

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.

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.

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

Frequency table

What statistical information do tests measuring central tendency tell us?

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

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

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

Mean (M = x).

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