Standard Error Formula

Standard error pertains to the standard deviation of a data set, which is an essential statistical metric. This formula may be used to determine the accuracy of a sample that reflects a population. The standard error formula is the discrepancy between the sample mean and the population mean.

Standard Error

The term “sample” in statistics refers to a specific set of information that is generated. The data we obtained on the height of some people in a locality, for example, maybe the sample. A population is a collection of people from which we draw a sample. There are several ways to define a population, and we must always be clear about what constitutes a population. This collection necessitates a large number of calculations.

Standard error represents how well a given sample represents the population. The Standard Error indicates how well the sample mean predicts the real population mean.

Formula

The SE formula is used to determine the reliability of a sampling that represents a population. The sample mean that differs from the provided population and is expressed as:

SE = 

where,

• S denotes standard deviation of the data
• n denotes the number of observations

Sample Problems

Question 1: Find the standard error for the sample data: 1, 2, 3, 4, 5.

Solution:

Mean of the given data = (1+2+3+4+5)/5

= 15/5

= 3

Standard Deviation = √((1 – 3)² + (2 – 3)² + (3 – 3)² + (4 – 3)² + (5 – 3)²)/(5 – 1)

= √((4 + 1 + 0 + 1 + 4)/4)

= √(10/4)

= 1.5

Now, SE = 1.5/√5

= 0.67

Question 2: Find the standard error for the sample data: 2, 3, 4, 5, 6.

Solution:

Mean of the given data = (2+3+4+5+6)/5

= 20/5

= 4

Standard Deviation = √((2 – 4)² + (3 – 4)² + (4 – 4)² + (5 – 4)² + (6 – 4)²)/(5 – 1)

= √((4 + 1 + 0 + 1 + 4)/4)

= √(10/4)

= 1.58

Now, SE = 1.58/√5

= 0.706

Question 3: Find the standard error for the sample data: 10, 20, 30, 40, 45.

Solution:

Mean of the given data = (10+20+30+40+50)/5

= 145/5

= 29

Standard Deviation = √((10 – 29)² + (20 – 29)² + (30 – 29)² + (40 – 29)² + (45 – 29)²)/(5 – 1)

= √(820/4)

= 14.317

Now, SE = 14.317/√6

= 5.84

Question 4: Find the standard error for the sample data: 2, 6, 9, 5.

Solution:

Mean of the given data = (2+6+9+5)/4

= 5.5

Standard Deviation = √((2 – 5.5)² + (6 – 5.5)² + (9 – 5.5)² + (5 – 5.5)²)/(4 – 1)

= √(25/3)

= 2.88

Now, SE = 2.8/√5.5

= 1.19

Question 5: Find the standard error for the sample data: 5, 8, 10, 12.

Solution:

Mean of the given data = (5+8+10+12)/4

= 8.75

Standard Deviation = √((5 – 8.75)² + (8 – 8.75)² + (10 – 8.75)² + (12 – 8.75)²)/(4 – 1)

= √(26.75/3)

= 2.98

Now, SE = 2.98/√8.75

= 1.0074

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