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### Course: Statistics and probability > Unit 3

Lesson 4: Variance and standard deviation of a population- Measures of spread: range, variance & standard deviation
- Variance of a population
- Population standard deviation
- The idea of spread and standard deviation
- Calculating standard deviation step by step
- Standard deviation of a population
- Mean and standard deviation versus median and IQR
- Concept check: Standard deviation
- Statistics: Alternate variance formulas

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# The idea of spread and standard deviation

See how distributions that are more spread out have a greater standard deviation.

## Introduction to standard deviation

Standard deviation measures the spread of a data distribution.

**The more spread out a data distribution is, the greater its standard deviation.**For example, the blue distribution on bottom has a greater standard deviation (SD) than the green distribution on top:

Interestingly, standard deviation cannot be negative. A standard deviation close to $0$ indicates that the data points tend to be close to the mean (shown by the dotted line). The further the data points are from the mean, the greater the standard deviation.

## Try it yourself

## Want to join the conversation?

- what made this so important for math(20 votes)
- Statistics is used for a lot of everyday things. While you may not personally calculate statistical values, statistics is important for business, sports, video games, politics, medicine, software, etc. No matter what field you go into, that field will use statistics in some way, shape, or form.(45 votes)

- Can be standard deviation be zero?

then what is its significance? can we say this is a statistical problem?(4 votes)- It can be zero if all entries have the same value. This is unlikely but possible to get such small sample from discrete distribution. Nevertheless, if you get big sample where each entry has exact the same value this should lead to the idea there is something wrong with the data source.(21 votes)

- why do I need to know this(8 votes)
- got this answer from the user screenbones: Statistics is used for a lot of everyday things. While you may not personally calculate statistical values, statistics is important for business, sports, video games, politics, medicine, software, etc. No matter what field you go into, that field will use statistics in some way, shape, or form.(9 votes)

- Can anyone please explain the difference for

Population Standard Deviation Vs Sample Standard Deviation?(4 votes)- Population Standard Deviation is used when you're taking ALL the data observed as a set.

Sample Standard Deviation is used when you're takin only a SUBSET of the data observed in a set.(13 votes)

- how was the standard deviation determined?(3 votes)
- For this exercise, you don't have to calculate the standard deviations. Just look at the graphs and visually compare the distributions. Which distribution seems to have a wider spread of data around the mean? That is, which distribution includes points that are further from the mean (represented by the dotted line)? That is the distribution with the higher standard deviation.

The next lesson has a step-by-step walk-through for calculating the standard deviation.(6 votes)

- What is the formula for standard deviation?(5 votes)
- Set A and Set B have the same mean. Set A has a larger standard deviation. Is it true that Set A is more likely to have larger values?(4 votes)
- Standard deviation just shows us the distance of data points from the mean. If a value is large enough,the mean shifts towards that value.

However in the case of a distribution, like the one above where the mean is fixed (same as Set B)

The S.D. is proportional to the number of data points close(er) to the mean and not the actual value of that data. Therefore it's not the value that is important but the [val]-[mean], or the deviation from the mean.

Conclusion: The the value of individual data points can be small or large, S.D. measures their distance only.(2 votes)

- How do you find the mode?(3 votes)
- most common number(4 votes)

- What can we infer from the data if we say that the data has huge variation or the data is spread out from the mean or the data has high std.deviation? What difference will it make in inference as opposed to the std.deviation being close to 0.(3 votes)
- Hi Vrisha,

When the data has a high SD (e.g., SD = 25), it means that the individual data points are spread out quite a bit.

On the other hand, if the data set has a smaller SD (e.g., SD = 0.3), you can infer that the data points are closer to the mean.(3 votes)

- If Data Spread is high is good or bed ?(3 votes)
- Depends on the situation, and mean. For example, let's take a movie's score. Let's say Marvel says it is a 4.5/5 movie.You would want a low MAD. But, if the score is 1/5, you would want a high MAD, like 4.(2 votes)