Name: measuring data spread with range and variance
Uploaded: 2020-02-12T11:07:41.000Z
Duration: 2 min 32 s
Channel: JoVE

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Variation: Normal Distribution, Range, and Standard Deviation

During a data collection project, a student is interested in gathering the heights of adult men in their city.

Upon returning to the classroom, the pupils graph the frequency of heights in the sample population. The resulting curve is bell-shaped, with a single peak at the center of which lies the mean.

While a single data point, such as the mean, is crucial to the analysis of these results, so too is the variation. Defined as the dispersion of measurements within a data set, this quantity describes the spread of results, giving a sense of the distance between points.

Additionally, the graph is symmetric, with half of the individuals demonstrating a stature taller than and half shorter than the average. This is referred to as a normal distribution or curve.

To evaluate the variation, they first calculate the range of the results, which is the difference between the highest and lowest heights.

Although the range describes the spread of data, it can be dramatically affected by outliers&mdash;like the school&rsquo;s tallest basketball player&mdash;and doesn&rsquo;t elucidate how measurements are positioned around the mean.

To address this, the student uses an equation to compute a second gauge of variation,

variation: normal distribution, range, and standard deviation

In the field of psychology, there are&nbsp;several ways to organize measurements of a trait, feature, or characteristic (i.e., variables). Qualitative data, such as ethnicity, can be tabulated into a frequency count to provide information about the proportion, as well as the variety of groups in a sample or population. On the other hand, researchers can perform a wider set of calculations on quantitative data. The mean, mode, and median, for instance, are central tendency measures to identify a typical value of a variable within a given numerical data set. Likewise, there are also a few approaches to estimating the distance of scores from each other, referred to as variability or variation, including range, variance, and standard deviation.

Range 

The range&nbsp;calculates the distance or difference between only the highest and lowest scores of a variable but provides no details about the scores in between. A high value denotes a wider spread of scores, but outliers may result in misinterpretations. For these reasons, the range is considered a less precise method to measure variation.

Variance 

Researchers typically use variance to estimate the average distance of all scores in the sample or population around the mean. First, the mean is determined by dividing the sum of all the raw scores of a specific variable by the total number of scores in the sample. Subtracting the mean from each of the raw scores produces a set of deviation scores that will comprise of both positive as well as negative integers, depending on whether the scores are higher or lower than the mean. Attempting to compute the mean of deviation scores will be insufficient, because the positive integers will cancel negative integers leading to a sum of zero. Squaring the deviations converts the negative deviation scores into positive scores, while still providing a reasonable estimate of the distance between the mean and each data point. Totaling the squared deviations forms the sum of squares (SS).

The SS is divided by either the total number of data points (N) or degrees of freedom (N-1), if the variance is computed for a sample or estimated for a population of scores, respectively. Dividing the sum of squared deviations provides an aggregate estimate of the general distance between the scores and the mean.

Standard Deviation 

The square root of the variance is the standard deviation. This arithmetic step serves to counterbalance the squaring of deviations in the preceding step of the variance formula. The standard deviation not only describes the general spread of scores in a population or sample set, but it is also used to assess the distance between a particular score from the mean. If the scores follow a normal curve, the location of the score relative to the center of the curve can relate to its likelihood of occurrence (probability).

Implications of Variation 

A reduced range of scores in a sample or population corresponds to a decrease in variance. For example, data from females exhibit a low spread in characteristics such as verbal performance, math performance, and height when compared with males. In these cases, understanding sources of the variation among males, such as environmental or biological factors, is as important as recognizing between-group differences.

In the field of psychology, there are&nbsp;several ways to organize measurements of a trait, feature, or characteristic (i.e., variables). Qualitative ...

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measuring data spread with range and variance

Variation describes how far scores are from each other in a data set. In psychology, researchers organize measurements of traits, features, or characteristics, called variables, in different ways. Qualitative data, such as ethnicity, can be counted to show the proportion and variety of groups in a sample or population. Quantitative data allow for more calculations, including measures of central tendency and variability.

The mean, mode, and median are central tendency measures. They identify a typical value in a numerical data set. Variability, also called variation, describes the distance between scores. Range, variance, and standard deviation are three common ways to measure that spread.

Range is the simplest measure of variation. It uses only the highest and lowest scores and finds the distance between them. Range does not show anything about the scores in between. A high range suggests a wider spread, but outliers can make it misleading. For that reason, range is a less precise measure of variation.

Variance gives a better estimate of spread. Researchers first find the mean by dividing the sum of the raw scores by the number of scores. Then they subtract the mean from each score to make deviation scores. These deviations can be positive or negative, depending on whether a score is above or below the mean. If you average the deviation scores, the positives and negatives cancel out and the result is zero.

To avoid that problem, each deviation is squared. Squaring turns negative deviations into positive values and still keeps the distance from the mean. Adding the squared deviations gives the sum of squares, or SS. The SS is divided by N for a population or by N-1 for a sample. The result gives an overall estimate of how far scores are from the mean.

Standard deviation is the square root of the variance. This step reverses the effect of squaring the deviations. Standard deviation shows the general spread of scores in a sample or population. It also helps show how far one score is from the mean. If scores follow a normal curve, a score’s position near the center or far from it can relate to how likely that score is to occur.

A smaller range usually means a smaller variance. In the transcript, female data show a low spread in verbal performance, math performance, and height compared with male data. In these cases, it is important to think about why variation differs among males as well. Environmental and biological factors can both be part of the explanation.

Measuring Data Spread with Range and Variance

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Variation describes how far scores are from each other in a data set. In psychology, researchers organize measurements of traits, features, or characteristics, called variables, in different ways. Qualitative data, such as ethnicity, can be counted to show the proportion and variety of groups in a sample or population. Quantitative data allow for more calculations, including measures of central tendency and variability. The mean, mode, and median are central tendency measures. They identify a...

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Learn how range, variance, and standard deviation measure data spread, why outliers matter, and how normal curves relate to score likelihood.

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HS-LS3-2

Measuring Data Spread with Range and Variance

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