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Descriptive Stats: Central Tendency, Variability, and Normal Curve, Study notes of Research Methods in Psychology

An introduction to descriptive statistics, focusing on measures of central tendency (mean, median, mode), variability (range, variance, standard deviation, skew), and the normal probability curve. Descriptive statistics help organize, present, and summarize numerical data, revealing basic information and relationships between variables. Graphical methods, such as histograms and scatter plots, enhance understanding.

What you will learn

  • How does the normal probability curve represent a model distribution?
  • What are the different measures of central tendency and how do they differ?
  • What is the role of measures of variability in understanding data?

Typology: Study notes

2021/2022

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Descriptive Statistics
Introduction
Tests, experiments and studies in psychology provide valuable information, most of
the time in numerical scores, known as data. Data in its raw form has little meaning for the
reader and in order to be able to understand and interpret the data it needs to be arranged in a
systematic way. Descriptive statistics involves describing, organizing, presenting and
summarizing data via numerical calculations, graphs or tables. Descriptive statistics is useful
for two reasons- first, it provides basic information about a particular population or data set
and second, it helps highlight possible relationships between variables.
Graphical Representation of Data
Graphical or pictorial methods provide a visual representation of data, therefore
enhancing a researcher’s understanding of variables and the relationship between them. There
are various different methods for the graphical representation of data which include
histograms, frequency distributions, scatter plots, contingency tables etc.
Measures of Central Tendency
The most basic yet one of the most important and informative aspect of descriptive
statistics are the Measures of Central Tendency. They provide information regarding the
“average” of the population being studied. Measure of central tendency may be defined as a
sort of average or typical value of items in the series and its function is to summarize the
series in terms of this average value (Tate, 1955).
First, the mean also known as the arithmetic mean may be considered the most useful
measure of central tendency, it is the average computed. That is, the sum of all the variables
values divided by the number of values.
Second, the median is the middle value in a set of values arranged in and ascending or
descending series. Therefore, the median of distribution is the point on the scale below which
half of the scores fall. The central item itself is not the median but rather, it is the measure or
value of the central item.
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Descriptive Statistics Introduction Tests, experiments and studies in psychology provide valuable information, most of the time in numerical scores, known as data. Data in its raw form has little meaning for the reader and in order to be able to understand and interpret the data it needs to be arranged in a systematic way. Descriptive statistics involves describing, organizing, presenting and summarizing data via numerical calculations, graphs or tables. Descriptive statistics is useful for two reasons- first, it provides basic information about a particular population or data set and second, it helps highlight possible relationships between variables. Graphical Representation of Data Graphical or pictorial methods provide a visual representation of data, therefore enhancing a researcher’s understanding of variables and the relationship between them. There are various different methods for the graphical representation of data which include histograms, frequency distributions, scatter plots, contingency tables etc. Measures of Central Tendency The most basic yet one of the most important and informative aspect of descriptive statistics are the Measures of Central Tendency. They provide information regarding the “average” of the population being studied. Measure of central tendency may be defined as a sort of average or typical value of items in the series and its function is to summarize the series in terms of this average value (Tate, 1955). First, the mean also known as the arithmetic mean may be considered the most useful measure of central tendency, it is the average computed. That is, the sum of all the variables values divided by the number of values. Second, the median is the middle value in a set of values arranged in and ascending or descending series. Therefore, the median of distribution is the point on the scale below which half of the scores fall. The central item itself is not the median but rather, it is the measure or value of the central item.

Third, the mode is the value that occurs the most in a data set. It is the value in a series which is the most characteristic or common and is usually repeated the maximum number of times. Measures of Variability Data tends to be dispersed, scattered or to show variability around the average or the central value, this is known as dispersion or variability. Measures of variability also known as measures of dispersion, provide information about the spread of a variable’s values.

  • Range is the simplest but rough measure of variability shows the difference between the smallest value and the largest value in a data set.
  • Variance is the most commonly used measure of variability, calculated by taking average of the squared differences between each value and the mean.
  • Standard deviation is the square root of the arithmetic average of the squared deviations of scores from mean of the distribution. It is the most stable and reliable measure of variability.
  • Skew is a measure to find whether some values of variable are different from the majority of the values. There can be positive skew, where the extreme values are higher than the majority of values and negative skew where a variable is negatively skewed if the extreme values are lower than the majority of values. Percentile and Percentile Rank Unlike pure science, measurements in the field of psychology do not have absolute zeros. Therefore, interpretations can be made only when the performance of an individual is compared in relation to a reference group. Statistics like quartiles, deciles, percentiles, etc. show fixed points like median, on the score scale below which a definite proportion of cases lie.

Normal Probability Curve Probability is the measure of the likelihood of an event happening and is quantified as a number between 0 and 1 wherein 0 indicates no possibility and 1 implies certainty. The higher the probability of an event, the greater the certainty of the event occurring. A Normal Probability Curve, is a bell-shaped curve which shows the probability distribution of a continuous random variable and represents a normal distribution. The total area of the Normal Probability Curve represents the sum of all the probabilities for a random variable. The Normal Probability Curve is symmetrical, with the highest frequency in the middle and frequencies tapering off as one moves towards either extreme. In other words, the Normal Probability Curve has a single mode in the middle with the frequency decreasing as one moves away from the middle in either direction. The Normal Probability Curve was derived independently by Gauss and Laplace (1777-1855). It was also named the ‘curve of error’, where ‘error’ is used to indicate a deviation from the normal or the true value. The normal probability curve takes into account the law which states that the greater the deviation from the mean score or average score, the less frequently it occurs. Characteristics of Normal Probability Curve The normal probability curve is symmetrical. The Normal Probability Curve is perfectly symmetrical and is not skewed. This implies that the size, shape and slope of the curve on the left side is identical to that of the right side. The curve has bilateral symmetry. Half of the data will fall to the left of the mean while the other half will fall to the right. The normal probability curve is unimodal. There is only one point in the curve which has maximum frequency, hence the curve has only one mode and is unimodal. Mean, median and mode are the same. The mean, median and mode of the NPC are the same and lie at the centre. They are indicated by 0 along the base line. The Normal Probability Curve is asymptotic to the X-axis. The Normal Probability Curve approaches the X axis asymptotically but it never touches the X axis. That is, the curve decreases in height on both sides away from the midpoint. This is because there is a possibility of locating a case in the population that scores higher than the highest score or lower than the lowest score. Therefore, it extends from minus infinity to plus infinity.

Normal Probability Curve represents a model distribution and hence is particularly significant in social sciences and behavioural sciences. It can be used as a model to compare various distributions and determine whether the distribution is normal or not and to what extent it deviates from normality. References King, B.M., Minium, E.W, & Rosopa, P.J. (2018). Statistical Reasoning in the Behavioural Sciences (Seventh Edition). John Wiley & Sons , Inc.