The **sampling error calculator** is used to estimate the potential **error** or **deviation** between a **sample statistic** and the **true population parameter** (such as the population mean).

Sampling erroroccurs because the sample may not perfectly represent the entire population, leading toinherent variabilityin the data.

The **sampling error calculator** requires the following **inputs**:

**Sample Size**: The number of observations or data points in the sample.**Population Standard Deviation**: The known or estimated standard deviation of the population.**Confidence Level**: The desired level of confidence, typically**90%**,**95%**, or**99%**.

## Sampling Error Calculation Chart

Sample Size | Population Standard Deviation | 90% Confidence | 95% Confidence | 99% Confidence |
---|---|---|---|---|

30 | 2 | 0.69 | 0.92 | 1.29 |

30 | 3 | 1.04 | 1.38 | 1.94 |

30 | 4 | 1.38 | 1.84 | 2.58 |

50 | 2 | 0.49 | 0.65 | 0.91 |

50 | 3 | 0.74 | 0.98 | 1.37 |

50 | 4 | 0.98 | 1.31 | 1.84 |

100 | 2 | 0.35 | 0.46 | 0.64 |

100 | 3 | 0.52 | 0.69 | 0.97 |

100 | 4 | 0.69 | 0.92 | 1.30 |

# Sampling Error Formula

The formula for calculating the **sampling error** is:

**Sampling Error** = **Z** * (Population Standard Deviation / √Sample Size)

Where:

Z: Thez-scorecorresponding to the desired confidence level (e.g.,1.65for90%,1.96for95%,2.58for99%)

Population Standard Deviation: The known or estimated standard deviation of the population

Sample Size: The number of observations or data points in the sample

### Example

Let’s use the example where the researcher wants to estimate the mean height of all students in a university.

Given:

**Sample Size**: 50 students**Population Standard Deviation**: 3 inches**Confidence Level**: 95%

To calculate the **sampling error**, we need to find the corresponding **z-score** for a **95% confidence level**, which is **1.96**.

Plugging the values into the formula:

**Sampling Error** = **Z** *(Population Standard Deviation / √Sample Size)*

**Sampling Error** = **1.96** (3 / √50)

**Sampling Error** = **0.84 inches**

This means that the **true population mean height** is likely to be within **68 ± 0.84 inches**, or between **67.16 inches** and **68.84 inches**, with a **95% confidence level**.

The **sampling error** of **0.84 inches** represents the potential **deviation** or **margin of error** in the estimate of the population mean, due to the inherent variability in the sample data and the fact that the sample may not perfectly represent the entire population.

## How Do You Calculate Sampling Error?

To calculate the **sampling error**, you can follow these steps:

- Choose the desired level of confidence, typically
**90%**,**95%**, or**99%**. - Find the corresponding
**z-score**for the chosen confidence level. For example, the**z-score**for a**95% confidence level**is**1.96**. - Obtain the following information:
**Sample Size**(n)**Population Standard Deviation**(σ)

- Input the values into the
**sampling error formula**:**Sampling Error**=**Z*** (Population Standard Deviation / √Sample Size) Where:**Z**: The**z-score**corresponding to the desired confidence level**Population Standard Deviation**: The known or estimated standard deviation of the population**Sample Size**: The number of observations or data points in the sample

Suppose a **market research firm** wants to estimate the average monthly spending of all customers in a particular region.

They take a **random sample** of **75 customers** and find the sample mean monthly spending to be **$250**, with a known population standard deviation of **$50**.

The firm wants to calculate the **sampling error** at a **90% confidence level**.

Confidence Level: 90%

Z-score: For a90% confidence level, thez-scoreis1.65.

Sample Size: 75 customers

Population Standard Deviation: $50

Plugging the values into the **sampling error formula**:

**Sampling Error** = **Z** *(Population Standard Deviation / √Sample Size)*

**Sampling Error** = **1.65** ($50 / √75)

**Sampling Error** = **$12.03**

This means that the

true population mean monthly spendingis likely to be within$250 ± $12.03, or between$237.97and$262.03, with a90% confidence level.

The **sampling error** of **$12.03** represents the potential **deviation** or **margin of error** in the estimate of the population mean, due to the inherent variability in the sample data and the fact that the sample may not perfectly represent the entire population.