# Interpreting mean and standard deviation results

The mean is the average. Add up the values, and divide by the number of values. The standard deviation SD quantifies variability. It is expressed in the same units as the data. It is often abbreviated as s. Prism computes the SD using a denominator of n-1, so computes what is sometimes called the sample SD rather than the population SD.

It is a measure of how far your sample mean is likely to be from the true population mean. The variance equals the SD squared, and therefore is expressed in the units of the data squared. Mathematicians like to think about variances because they can partition variances into different components -- the basis of ANOVA. In contrast, it is not correct to partition the SD into components.

Because variance units are usually impossible to think about, most scientists avoid reporting the variance of data, and stick to standard deviations. Prism does not report the variance. All rights reserved. This guide is for an old version of Prism. Browse the latest version or update Prism. Mean The mean is the average. Standard Deviation The standard deviation SD quantifies variability.

Variance The variance equals the SD squared, and therefore is expressed in the units of the data squared. Scroll Prev Top Next More.A boxplot provides a graphical summary of the distribution of a sample. The boxplot shows the shape, central tendency, and variability of the data. Use a boxplot to examine the spread of the data and to identify any potential outliers.

Boxplots are best when the sample size is greater than Examine the shape of your data to determine whether your data appear to be skewed. When data are skewed, the majority of the data are located on the high or low side of the graph. Often, skewness is easiest to detect with a histogram or boxplot. The boxplot with right-skewed data shows wait times. Most of the wait times are relatively short, and only a few wait times are long.

Mean and Standard Deviation; Computation and Interpretation

The boxplot with left-skewed data shows failure time data. A few items fail immediately, and many more items fail later. Try to identify the cause of any outliers. Correct any data—entry errors or measurement errors.

Consider removing data values for abnormal, one-time events also called special causes. Then, repeat the analysis. For more information, go to Identifying outliers.

The coefficient of variation CoefVar is a measure of spread that describes the variation in the data relative to the mean. The coefficient of variation is adjusted so that the values are on a unitless scale.

Because of this adjustment, you can use the coefficient of variation instead of the standard deviation to compare the variation in data that have different units or that have very different means. For this ordered data, the first quartile Q1 is 9. A histogram divides sample values into many intervals and represents the frequency of data values in each interval with a bar. Use a histogram to assess the shape and spread of the data.

Histograms are best when the sample size is greater than You can use a histogram of the data overlaid with a normal curve to examine the normality of your data. A normal distribution is symmetric and bell-shaped, as indicated by the curve. It is often difficult to evaluate normality with small samples.

A probability plot is best for determining the distribution fit. An individual value plot displays the individual values in the sample. Each circle represents one observation. An individual value plot is especially useful when you have relatively few observations and when you also need to assess the effect of each observation. Use an individual value plot to examine the spread of the data and to identify any potential outliers.

Individual value plots are best when the sample size is less than The individual value plot with right-skewed data shows wait times. The individual value plot with left-skewed data shows failure time data.

Outliers, which are data values that are far away from other data values, can strongly affect the results of your analysis. Often, outliers are easiest to identify on a boxplot.Closed ended rating scale data is easy to summarize and hard to interpret. Ideally you can compare the responses to an industry benchmark, a competitor or even a similar survey question from a prior survey. For example, a recent survey I worked on asked a question about what users thought of the visual appeal of the software.

Users were given a five point rating scale from strongly disagree to strongly agree. To find more meaning in this jumble of numbers, the first thing you need to do is compute the mean and standard deviation. There were 18 responses and the mean was a 4. Here are five ways of making the raw responses more interpretable. As you can see, many of the methods generate reassuringly similar results. However, there are times when executive comprehension is more important than statistical precision. If you find it hard to explain the z-score approach and are unsure whether others will be comfortable with it, one of the other approaches will generate similar results albeit less precisely.

To help you get started, you can download an Excel file with the appropriate calculations for 5 and 7 point scales. This leaves product managers and researchers to do their best in interpreting the raw responses.

The top-two box score is the same as the agree score. The popular Net Promoter Score uses a variation on this one it subtracts the bottom six from the top 2 boxes.

A Forrester annual report called the Customer Experience Index subtracts the top 2 bottom responses from the top-2 top responses called the CxPi. It converts the raw score into a normal score—because rating scale means often follow a normal or close to normal distribution. We just need a reasonable benchmark to compare the mean to. Next follow these three steps. Subtract the benchmark from the mean: 4. This is called a z-score or normal score and tells us how many standard deviations a score of 4. Convert the Z-score to a percentile rank: Using the properties of the normal curve we find out what percent of area falls below the. The CV makes interpreting a bit easier by dividing the standard deviation by the mean 1.

Higher values indicate higher variability. The CV is a measure of variability, unlike the first four which are measures of the central tendency, so it can be used in addition to the other approaches. It offers the most precision because it uses the mean.Have you ever had to explain to anyone what a standard deviation is? Or perhaps, you aren't sure yourself what it is.

What is it used for? Why is it important in process improvement? When using control charts, the standard deviation, as well as the average, is a very important parameter. One must understand what is meant by the standard deviation. This newsletter addresses this. We will start with describing what an average is. The average also called the mean is probably well understood by most.

It represents a "typical" value. For example, the average temperature for the day based on the past is often given on weather reports. It represents a typical temperature for the time of year. The average is calculated by adding up the results you have and dividing by the number of results. For example, suppose we have wire cable that is cut to different lengths for a customer.

These lengths, in feet, are 5, 6, 2, 3, and 8. The average is determined by adding up these five numbers and dividing by 5. In this case, the average X is:. The average length of wire for these five pieces is 4. While the average is understood by most, the standard deviation is understood by few.

To begin to understand what a standard deviation is, consider the two histograms. Histogram 1 has more variation than Histogram 2. In the first histogram, the largest value is 9, while the smallest value is 1.

The range is larger for Histogram 1. Ranges are often used in control charts for variation for example, the X -R charts. In fact, the average range from a control chart can be used to calculate the process standard deviation.

The average of the data in each histogram is 5. So, in this case, the highest bar is the average. We can also see that Histogram 1 has more variation than Histogram 2 because the distance, on average, of the individual observations from the overall average 5 is greater in Histogram 1 than Histogram 2.

## Standard Deviation and Variance

This distance is usually referred to as a deviation. One can view the standard deviation as being an "average" distance each individual measurement is from the average, X. Let's return to the numbers we found the average for earlier to see how we can estimate this average deviation from X.

These numbers were the length of wire cable we had cut. To do this, we can determine the deviation of each number from the average as shown below.In statistics, the range is a measure of the total spread of values in a quantitative dataset. Unlike other more popular measures of dispersion, the range actually measures total dispersion between the smallest and largest values rather than relative dispersion around a measure of central tendency.

The range is interpreted as t he overall dispersion of values in a dataset or, more literally, as the difference between the largest and the smallest value in a dataset. The range is measured in the same units as the variable of reference and, thus, has a direct interpretation as such.

This can be useful when comparing similar variables but of little use when comparing variables measured in different units. However, because the information the range provides is rather limited, it is seldom used in statistical analyses. For example, if you read that the age range of two groups of students is 3 in one group and 7 in another, then you know that the second group is more spread out there is a difference of seven years between the youngest and the oldest student than the first which only sports a difference of three years between the youngest and the oldest student.

The mid-range of a set of statistical data values is the arithmetic mean of the maximum and minimum values in a data set, defined as:. The mid-range is the midpoint of the range; as such, it is a measure of central tendency.

The mid-range is rarely used in practical statistical analysis, as it lacks efficiency as an estimator for most distributions of interest because it ignores all intermediate points. The mid-range also lacks robustness, as outliers change it significantly. Indeed, it is one of the least efficient and least robust statistics. Variance is the sum of the probabilities that various outcomes will occur multiplied by the squared deviations from the average of the random variable.

When describing data, it is helpful and in some cases necessary to determine the spread of a distribution. In describing a complete population, the data represents all the elements of the population. When determining the spread of the population, we want to know a measure of the possible distances between the data and the population mean. These distances are known as deviations. The variance of a data set measures the average square of these deviations.

More specifically, the variance is the sum of the probabilities that various outcomes will occur multiplied by the squared deviations from the average of the random variable. When trying to determine the risk associated with a given set of options, the variance is a very useful tool. Calculating the variance begins with finding the mean. Once the mean is known, the variance is calculated by finding the average squared deviation of each number in the sample from the mean. For the numbers 1, 2, 3, 4, and 5, the mean is 3. The calculation for finding the mean is as follows:.You can manage your email preferences or unsubscribe at any time.

Standard Deviation and Standard Error are perhaps the two least understood statistics commonly shown in data tables.

### How to Find the Mean, Median, Mode, Range, and Standard Deviation

The following article is intended to explain their meaning and provide additional insight on how they are used in data analysis. Both statistics are typically shown with the mean of a variable, and in a sense, they both speak about the mean.

They are often referred to as the "standard deviation of the mean" and the "standard error of the mean. Standard Deviation often abbreviated as "Std Dev" or "SD" provides an indication of how far the individual responses to a question vary or "deviate" from the mean. Did all of your respondents rate your product in the middle of your scale, or did some love it and some hate it?

Let's say you've asked respondents to rate your product on a series of attributes on a 5-point scale. The mean for a group of ten respondents labeled 'A' through 'J' below for "good value for the money" was 3. At first glance looking at the means only it would seem that reliability was rated higher than value.

But the higher SD for reliability could indicate as shown in the distribution below that responses were very polarized, where most respondents had no reliability issues rated the attribute a "5"but a smaller, but important segment of respondents, had a reliability problem and rated the attribute "1". Looking at the mean alone tells only part of the story, yet all too often, this is what researchers focus on.

The distribution of responses is important to consider and the SD provides a valuable descriptive measure of this. Two very different distributions of responses to a 5-point rating scale can yield the same mean.

Consider the following example showing response values for two different ratings. The individual responses did not deviate at all from the mean. In Rating "B", even though the group mean is the same 3. The Standard Deviation of 1. Another way of looking at Standard Deviation is by plotting the distribution as a histogram of responses. A distribution with a low SD would display as a tall narrow shape, while a large SD would be indicated by a wider shape. SD generally does not indicate "right or wrong" or "better or worse" -- a lower SD is not necessarily more desireable.

It is used purely as a descriptive statistic. It describes the distribution in relation to the mean. However, it is not actually calculated as an average if it were, we would call it the "average deviation".

Instead, it is "standardized," a somewhat complex method of computing the value using the sum of the squares. For practical purposes, the computation is not important. Most tabulation programs, spreadsheets or other data management tools will calculate the SD for you.

More important is to understand what the statistics convey. A small SE is an indication that the sample mean is a more accurate reflection of the actual population mean. A larger sample size will normally result in a smaller SE while SD is not directly affected by sample size. Most survey research involves drawing a sample from a population. We then make inferences about the population from the results obtained from that sample. If a second sample was drawn, the results probably won't exactly match the first sample.

If the mean value for a rating attribute was 3.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization.

It only takes a minute to sign up. I recently presented a national test and the company in charge of preparing the test then does a standardization to provide the final scores for each person.

These are the values they gave at the end of the page where the grades were posted:. I don't remember anything about this, and my career doesn't deal with any of this.

I want to know how can I interpret these results with those values. They had a test. The results of that test were measurements. When they average those measurements, the value they come up with is When they compute the standard deviation using the data and the mean, the number they came up with is 7. So far, this is nearly uninformative. Mean is meant "measure of central tendency".

If you are going to bet, and you want to win, if you bet on the average value, then most likely you are going to do better than you would picking any other value. Standard deviation is a "measure of dispersive tendency". It is how wide a range the values span.

It is the "turning radius" of the data - does it take miles, or 1 inch. A smaller stdev means the variation is small. A large stdev means the variation is large. But there are a lot of assumptions here, and they aren't stated. If the tests was mean body weight between Sudan vs.