Measures of Spread

Measures of spread are ways of summarizing a group of data by describing how spread out the values are.  If the spread of values in the data set is large, the mean is not as representative of the data as if the spread of data is small. This is because a large spread indicates that there is a lot of variation between individual data points.​

The spread of the values can be measured for quantitative data, as the variables are numeric and can be arranged into a logical order with a low end value and a high end value. Measures of spread are used in conjunction with a measure of central tendency, such as the mean or median, to provide an overall description of a set of data. To describe spread, a number of statistics are available, including the range, quartiles and standard deviation. Which you use will depend on how much and the type of data you collected.

What Measure of Central Tendency Did you Calculate?
Mean with 5 or more trials per level of manipulation	Standard Deviation
Mean with 4 or less trials per level of manipulation	Range
Median with 5 or more trials per level of manipulation	Quartile
Median with 4 or less trials per level of manipulation	Range

Standard Deviation

Averages do not tell us everything about a sample.  Samples can be very uniform with the data all bunched around the mean (Figure 1) or they can be spread out a long way from the mean (Figure 2). The statistic that measures this spread for normally distributed data is called the standard deviation.  The standard deviation is average distance each data point is from the sample mean.  The wider the spread of scores, the larger the standard deviation. 

For data that has a normal distribution, 68% of the data lies within one standard deviation of the mean.

How to Calculate the Standard Deviation:  ​

The formula for calculating the standard deviation of a sample can be solved by following these steps: 

1. Calculate the mean (x̅) of the sample data​
2. Find the difference of each point of data from the mean.  You'll do this for each data point, so you'll have multiple (x-x̅). 
3. Square each of the resulting differences to determine (x-x̅)^2.  As in step 2, you'll do this for each data point, so you'll have multiple (x-x̅)^2.   This step removes any negative values.
4. Add the squared differences from the previous step together to get ∑(x-x̅)^2.  Now you should be working with a single value, called the sum of squares.  
5. Calculate the degrees of freedom, (n-1), by subtracting 1 from the sample size.  
6. Divide the sum of squares (answer from step 4) by the degrees of freedom (answer from step 5).  This gives the sample variance.
7. Take the square root of the sample variable (answer from step 6) to determine the sample standard deviation.
8. Be sure your standard deviation has the same number of digits as your raw data, so you may need to round your answer.  
9. The standard deviation should have the same unit as the raw data you collected.  For example, SD = +/- 0.5 cm.  

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Calculating the Standard Deviation in Spreadsheets

• The formula is  =STDEV.S(A1:A100).  Replace A1:A100 with your data by highlighting the cells with data you want to include in the standard deviation calculation.
• Make sure that you set the values to the correct number of digits.  Excel and Sheets with both default to returning too many numbers after the decimal place.  Your mean and standard deviation must have the same precision (number of digits after the decimal) as your data points.  

Quartiles

Quartiles divide an ordered dataset into four equal parts, and refer to the values of the point between the quarters. Quartiles are a useful measure of spread because they are much less affected by outliers or a skewed data set than the standard deviation. For this reason, quartiles are often reported along with the median as the best choice of measure of spread and central tendency, respectively, when dealing with skewed and/or data with outliers. 

A common way of expressing quartiles is as an interquartile range. The interquartile range (IQR) describes the difference between the third quartile (Q3) and the first quartile (Q1), telling us about the range of the middle half of the scores in the distribution.  The IQR is often seen as a better measure of spread than the range as it is not affected by outliers

• Online calculators, such as https://www.calculatorsoup.com/calculators/statistics/quartile-calculator.php 
  
• Excel or Google Sheets =QUARTILE(highlight data cells,quart)
  
  • quart - The quartile value to return.  1 is the first quartile, 2 is the medium, 3 is the third quartile.
    
  • Example =QUARTILE(A1:A20,3)
    

Range

Range is the difference between the smallest value and the largest value in a dataset.  Range should only be used if there are less than 5 trials that are being used to calculate a measure of central tendency.   

A larger range indicates a larger spread of the data.  A larger range indicates that the data has greater variability.

• Online calculators, such as http://www.alcula.com/calculators/statistics/range/ 
• ​Excel or Google Sheets =MAX(highlight data cells)-MIN(highlight data cells)