Learning Objectives<\/h2>\n<\/header>\n\n\nBy the end of this section, you will be able to:\n\n \t- Understand descriptive statistics and know how to produce them.<\/li>\n \t
- Understand inferential statistics and why they are used.<\/li>\n<\/ul>\n<\/div>\n<\/div>\nOnce data is collected from the research participants, a set of statistical analyses is conducted to interpret the results. Descriptive statistics<\/strong> organise and summarise some important properties of the data set. Researchers usually describe the central tendency and variability of a data set; this allows them to quickly make some basic interpretations about the results of a large sample of people. They may also use frequency distributions and histograms to visualize the data set. Inferential statistics<\/strong> provide researchers with the tools to make inferences about the meaning of the results, specifically about generalising from the sample they used in their research to the greater population that the sample represents. In experimental studies, researchers run inferential statistics to determine the likelihood that the effect of treatment is due to chance (and thus not meaningful). Generally, psychologists consider differences to be statistically significant if there is less than a five percent chance of observing them if the groups did not actually differ from one another. Stated another way, psychologists want to limit the chances of making false-positive claims to five percent or less.\n
Descriptive Statistics<\/h1>\nLet\u2019s work through a hypothetical example to show how descriptive statistics help researchers to understand their data. Let\u2019s assume that we have asked 40 people to report how many hours of moderate-to-intense physical activity they get each week. Let\u2019s begin by constructing a frequency distribution of our hypothetical data that will show quickly and graphically what scores we have obtained.\n
- \n \t
- Understand descriptive statistics and know how to produce them.<\/li>\n \t
- Understand inferential statistics and why they are used.<\/li>\n<\/ul>\n<\/div>\n<\/div>\nOnce data is collected from the research participants, a set of statistical analyses is conducted to interpret the results. Descriptive statistics<\/strong> organise and summarise some important properties of the data set. Researchers usually describe the central tendency and variability of a data set; this allows them to quickly make some basic interpretations about the results of a large sample of people. They may also use frequency distributions and histograms to visualize the data set. Inferential statistics<\/strong> provide researchers with the tools to make inferences about the meaning of the results, specifically about generalising from the sample they used in their research to the greater population that the sample represents. In experimental studies, researchers run inferential statistics to determine the likelihood that the effect of treatment is due to chance (and thus not meaningful). Generally, psychologists consider differences to be statistically significant if there is less than a five percent chance of observing them if the groups did not actually differ from one another. Stated another way, psychologists want to limit the chances of making false-positive claims to five percent or less.\n
Descriptive Statistics<\/h1>\nLet\u2019s work through a hypothetical example to show how descriptive statistics help researchers to understand their data. Let\u2019s assume that we have asked 40 people to report how many hours of moderate-to-intense physical activity they get each week. Let\u2019s begin by constructing a frequency distribution of our hypothetical data that will show quickly and graphically what scores we have obtained.\n
![\"A](\"https:\/\/opentextbc.ca\/psychologymtdi\/wp-content\/uploads\/sites\/470\/2024\/02\/Research-method-2.png\")
Figure PS.13. Histogram.<\/strong> Histogram of exercise frequency.[\/caption]\n\nMany variables that interest psychologists have distributions where most of the scores are located near the centre of the distribution, the distribution is symmetrical, and it is bell-shaped. A data distribution that is shaped like a bell is known as a normal distribution<\/strong>. Normal distributions are common in human traits, such as intelligence, height, and shoe size. Relatively few people are either extremely high or low scorers, and most people fall somewhere near the middle.![\"This](\"https:\/\/opentextbc.ca\/psychologymtdi\/wp-content\/uploads\/sites\/470\/2024\/08\/Figure-3-5-768x321-4.jpg\")
Figure PS.14. Normal Distribution.<\/strong> The distribution of the heights of the students in a class will form a normal distribution. In this sample the mean (M) = 67.12 inches (170.48 cm) and the standard deviation (SD) = 2.74 inches (6.96 cm). ![\"This](\"https:\/\/opentextbc.ca\/psychologymtdi\/wp-content\/uploads\/sites\/470\/2024\/08\/Figure-3-6-768x441-3.jpg\")
Figure PS.15. Non-Symmetrical Distribution.<\/strong> The distribution of family incomes is likely to be non-symmetrical because some incomes can be very large in comparison to most incomes. In this case, the median or the mode is a better indicator of central tendency than is the mean. ![\"(a)](\"https:\/\/opentextbc.ca\/psychologymtdi\/wp-content\/uploads\/sites\/470\/2024\/08\/Bellshape.png\" "\\"(a)")
Figure PS.16. Low Variability and High Variability.<\/strong> When data are normally distributed, it is easy to see the difference between (a) a data set with low variability and (b) a data set with high variability.[\/caption]\n\nOne simple way to measure variability is to find the maximum score (i.e., the largest number in the data set) and the minimum score (i.e., the smallest number in the data set) and to compute the range <\/strong>of the variable as the maximum observed score minus the minimum observed score. In the previous example (Figure PS.14) , the range of height is 72 \u2013 62 = 10.\n\nThe standard deviation<\/strong>, or SD<\/em>, is the most commonly used measure of variability around the mean. Distributions with a larger standard deviation have more spread. Those with small deviations have scores that do not stray very far from the average score. Thus, standard deviation is a good measure of the average deviation from the mean in a set of scores. In the examples above, the SD<\/em> of height is 2.74, and the SD <\/em>of family income is 745,337. Standard deviation can be useful to compare the variability across samples. For example, a professor can keep track of student grades over many semesters. If the standard deviations are similar from semester to semester, this indicates that the amount of variability in student performance is fairly constant. A standard deviation that suddenly goes up indicates that there are more students with very low scores, very high scores, or both.\n\nThe standard deviation in the normal distribution has some interesting properties. Approximately 68% of the data fall within 1 standard deviation above or below the mean score: 34% fall above the mean, and 34% fall below. In other words, if a variable is normally distributed (e.g., height and IQ), approximately 2\/3 of the population are within 1 standard deviation around the mean. Likewise, the 2 standard deviations account for 95% of the population, and the 3 standard deviations include almost everyone (99.73%).\n\n[caption id=\"attachment_78\" align=\"aligncenter\" width=\"769\"]![\"This](\"https:\/\/opentextbc.ca\/psychologymtdi\/wp-content\/uploads\/sites\/470\/2024\/08\/Empirical_Rule-1.png\" "\\"This")
Figure PS.17. Standard Deviation.<\/strong> For the normal distribution, 68.27% of the scores are within 1 standard deviation from the mean; while 95.45% are within 2 standard deviations from the mean; and 3 standard deviations account for 99.73% of the scores.[\/caption]\n