**Sampling Error Formula (Table of Contents)**

## What is the Sampling Error Formula?

The term “Sampling Error” refers to the estimation error due to variance between the statistical characteristic of a population and a sample, which is the subset of the same population. In other words, it is the measure of the inaccuracy of the sample’s statistical characteristics from that of the population since the sample doesn’t include all members of the population. The higher value of sampling error indicates that the sample statistics are less representative of the actual population parameters. The formula for sampling error can be derived based on the confidence level of the estimation, sample size, population size and proportion of the population who are expected to respond in a certain way. Mathematically, it is represented as,

Formula

**Sampling Error = Z * √(p**

*** (1 – p**

**) / n] * [1 – √(n / N))**

Where,

**Z:**Z-Score for the Confidence Level Selected.**n:**Sample Size.**N:**Population Size.**p:**Proportion or Percentage of people surveyed who are Expected to respond in a certain way.

**Example of Sampling Error Formula (With Excel Template)**

Let’s take an example to understand the calculation of Sampling Error in a better manner.

#### Sampling Error Formula – Example #1

**Let us take the example of a sample of 500 people from an entire population of 100 million who were surveyed whether or not they like Vanilla ice creams. 70% of the sample responded positively saying that they like Vanilla ice creams. Calculate the sampling error for a 95% Confidence Level and 99% Confidence Level.**

**Solution:**

Sampling Error is calculated using the formula given below

**Sampling Error = Z * √(p**** * (1 – p****) / n) * (1 – √(n /N))**

z-score at 95%

- Sampling Error = 1.96 * √[70% * (1 – 70%) / 500] * [1 – √(500 / 100000000)]
- Sampling Error =
**4.01%**

Therefore, the sampling error for the sample at 95% confidence level is 4.01%.

z-score at 99%

- Sampling Error = 2.58 * √[70% * (1 – 70%) / 500] * [1 – √(500 / 100000000)]
- Sampling Error =
**5.28%**

Therefore, the sampling error for the sample at a 99% confidence level has gone up to 5.28%.

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Therefore, it can be observed that the sampling error of any sample increases with the increase in confidence level.

#### Sampling Error Formula – Example #2

**Now, again let us take the example the above example and keep everything the same except the population size which is to be assumed to be significantly lower in this case, say 2,000. Calculate the sampling error for a 95% confidence level and a 99% confidence level.**

**Solution:**

Sampling Error is calculated using the formula given below

**Sampling Error = Z * √(p**** * (1 – p****) / n) * (1 – √(n /N))**

z-score at 95%

- Sampling Error = 1.96 * √[70% * (1 – 70%) / 500] * [1 – √(500 / 2000)]
- Sampling Error =
**2.01%**

Therefore, the sampling error for the sample at 95% confidence level is 2.01%.

z-score at 99%

- Sampling Error = 2.58 * √[70% * (1 – 70%) / 500] * [1 – √(500 / 2000)]
- Sampling Error =
**2.64%**

Therefore, the sampling error for the sample at a 99% confidence level is 2.64%.

Therefore, it can be seen that the sampling error decreases with a decrease in population size. So, samples are a better representative of the smaller data population.

### Explanation

The formula for Sampling Error analysis can be computed by using the following steps:

**Step 1:** Firstly, decide on the confidence level to be used for the estimation. Based on the selected confidence level, the z-score can be determined that is denoted by “Z”. For instance, the z-score for a 95% confidence level is 1.96.

**Step 2:** Next, determine the sample size for the estimation. It is the proportion of the population that is expected to represent the entire population i.e. its sample characteristics will mostly be similar to that of the entire population. It is denoted by “n”.

**Step 3:** Next, determine the size of the entire population that is denoted by “N”.

**Step 4:** Next, determine the proportion of the people surveyed who are likely to respond either in a positive way or say “yes” as an answer to the survey question. It is expressed in percentage and denoted by “p”. So, (1 – p) denotes the percentage of the people with the alternate response.

**Step 5:** Final, the formula for sampling error can be derived based on the confidence level of the estimation (step 1), sample size (step 2), population size (step 3) and proportion of the population with a set response (step 4) as shown below.

**Sampling Error = Z * √(p * (1 – p) / n) * (1 – √(n /N))**

### Relevance and Use of Sampling Error Formula

It is very important to understand the concept of sampling error as it indicates the inaccuracy of the sample survey. A higher value of sampling error means that the survey may not be the true reflection of the population. On the other hand, a smaller value is desirable as it indicates that the sample parameters are close to that of the total population.

### Sampling Error Formula Calculator

You can use the following Sampling Error Formula Calculator

Z | |

p | |

n | |

N | |

Sampling Error | |

Sampling Error = | Z * √[p * (1 - p) / n] * [1 - √(n/N)] | |

0 * √[0 * (1 - 0) / 0] * [1 - √(0/0)] = | 0 |

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This is a guide to Sampling Error Formula. Here we discuss how to calculate the Sampling Error along with practical examples. We also provide a Sampling Error calculator with a downloadable excel template. You may also look at the following articles to learn more –