Systematic Random Sampling

Systematic random sampling selects samples systematically from a population based on a fixed interval.

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Systematic random sampling is a simpler alternative to simple random sampling, suitable for small, homogeneous populations with an ordered sampling frame.

Example:

• To estimate rice production per hectare in a region, the population consists of rice fields owned by residents.
• Fields are ordered by size (smallest to largest), and samples are drawn systematically.

Systematic random sampling involves selecting the first unit randomly, then following a fixed pattern for subsequent units.

Performing systematic random sampling

1. Determine the interval ($k$): $k=\frac{N}{n}$ Where:
   • $N$: Population size
   • $n$: Sample size
   • $k$: Sampling interval (number of possible samples or groups)
2. Randomly choose a unit $X_j$ where $j<k$.
3. Choose subsequent units: $X_{j+k},X_{j+2k},X_{j+3k},\dots$

Example:

• Population $N=15$, sample $n=3$, so $k=15/3=5$.
• If $X_1$ is chosen, the sample is $X_1,X_6,X_{11}$.

Characteristics

Compared to simple random sampling

• Simple random sampling: Samples are scattered randomly.
• Systematic random sampling: Samples follow a systematic pattern.

Conditions for use

1. A complete, up-to-date sampling frame exists.
2. The sampling frame follows a specific order (e.g., student IDs from smallest to largest).

Weaknesses

• If the sampling frame has systematic errors, bias may occur.
• Example: Sampling only husbands or wives from a list of 100 couples ($N=200$, $n=20$, $k=10$) could result in a non-representative sample (e.g., all husbands: $X_1,X_{11},X_{21},\dots$).

Solution to the weaknesses

• Reorder the sampling frame to ensure homogeneity (e.g., list all husbands first, then all wives).
• Example revised sample: $X_1$ (husband), $X_{11}$ (husband), $X_{91}$ (husband), $X_{101}$ (wife), $X_{111}$ (wife), $X_{191}$ (wife).

Advantages

1. Simpler and faster than simple random sampling.
2. Does not require a random number table.
3. Effective for small, homogeneous populations with an ordered frame.

Methods

Method A

• Randomly select one unit from the first $k$ units, then proceed with interval $k$.
• Example: $N=12$, $k=3$, possible samples:
  • $X_1,X_4,X_7,X_{10}$
  • $X_2,X_5,X_8,X_{11}$
  • $X_3,X_6,X_9,X_{12}$
• Probability of selecting any sample: $\frac{1}{k}$.

Method B

• Randomly select one unit from the entire population ($N$), then determine the starting point based on the remainder $r$ (where $r=j\mathrm{mod}k$).
• Example: $N=12$, $k=3$, if $X_{11}$ is chosen, $r=11\mathrm{mod}3=2$, start with $X_2$, yielding $X_2,X_5,X_8,X_{11}$.
• Probability of selecting a sample: $\frac{n}{N}$.

Considerations

• Systematic sampling assumes homogeneity within intervals. Consequently, it is less effective for heterogeneous populations or when $N$ is unknown (e.g., Method B cannot be used without a known $N$).
• If the population has hidden patterns (e.g., periodic trends), variance may increase, leading to bias.
• Precision depends on the sampling frame’s order and homogeneity.