Point Estimate of the Mean

A point estimate of the mean is a single numerical value used to approximate the average of a population.

Simply put, the point estimate of the mean is the average of a single data sample.

The term "point" signifies that it's a single number acting as an estimate, rather than an interval or range of values.

Note. This approach is useful when the entire population’s data is unavailable, as it would be too expensive or difficult to collect data from every individual.

How does it work?

It involves taking a representative sample and calculating the sample mean.

If you have a data sample with $n$ observations $x_1, x_2, \dots, x_n$, the point estimate of the population mean $ \mu $ is the sample mean $ \bar{x} $:

$$ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i $$

Where:

$ \bar{x} $ is the point estimate of the population mean,
$ n $ is the sample size (i.e., the number of units in the sample),
$ x_i $ are the individual observations in the sample.

This value serves as the "point estimate" of the population mean.

A practical example
Limitations of point estimation
Is one sample enough to estimate the population mean?
Additional notes

A practical example

Let’s say I want to estimate the average income of a population of 100 people but don’t have the resources to collect data from everyone.

Instead, I randomly select a sample of 5 people and obtain the following annual incomes (in thousands of euros):

$$ 40, 50, 60, 45, 55. $$

The point estimate of the population mean is the sample mean, or the average of the sample values:

$$ \bar{x} = \frac{40 + 50 + 60 + 45 + 55}{5} = \frac{250}{5} = 50 \text{ thousand euros}. $$

In this case, the point estimate of the population’s mean income is 50 thousand euros.

Limitations of point estimation

Point estimates don’t tell us anything about the accuracy or reliability of the estimate.

In other words, we can’t know how close the estimate is to the true population mean.

Furthermore, the point estimate might not be representative of the population, especially if the sample is small or not well-selected.

Note. To reduce this risk, it's generally better to work with larger samples, especially those with $ n \ge 30 $, for a few key reasons. Larger samples yield more precise and reliable estimates, allow the Central Limit Theorem to be applied, and reduce the impact of outliers and other biases. That said, if the population distribution is normal, even smaller samples can provide reliable estimates of the mean.

Why is it better to work with larger samples?

Central Limit Theorem (CLT)
The CLT tells us that if the sample size is sufficiently large (usually $ n \geq 30 $), the distribution of the sample means tends to be approximately normal, even if the population distribution is not. This allows us to apply statistical techniques based on normality, such as calculating confidence intervals and performing hypothesis tests.
Smaller standard error
The standard error of the sample mean is inversely proportional to the square root of the sample size, as shown in the formula: $$ e_x = \frac{\sigma}{\sqrt{n}} $$ where $ \sigma $ is the population standard deviation, and $ n $ is the sample size. As $ n $ increases, the standard error decreases, meaning the sample mean becomes a more accurate estimate of the true population mean. Simply put, a larger sample reduces variability in the sample means, making the estimates more reliable.
Less impact from outliers
In small samples, outliers can significantly skew the estimate of the mean. With larger samples, the influence of outliers is reduced, as extreme values are balanced by a greater number of central observations, making the sample mean more representative of the population.
Greater representativeness
Larger samples are typically more representative of the population. With a small sample, you might unintentionally select observations that don’t fully capture the diversity of the population. A larger sample is more likely to include a wider range of individuals, giving a better reflection of the entire population.
Narrower confidence intervals
Larger samples produce narrower confidence intervals, providing a more precise estimate. A narrow confidence interval indicates a high probability that the true population mean lies within a small range around the estimated mean.

For this reason, point estimates are often accompanied by confidence intervals, which provide a range of values within which we expect the true population mean to lie with a certain probability (for example, 95%).

An example of point estimation with a confidence interval

Let’s say I want to estimate the average annual income of employees at a company.

I can’t gather data from the entire population, so I randomly select a sample of 10 employees. Their annual incomes (in thousands of euros) are as follows:

$$ 35, 42, 39, 47, 41, 44, 38, 40, 36, 43 $$

The point estimate of the population mean is simply the arithmetic mean of the sample:

$$ \bar{x} = \frac{35 + 42 + 39 + 47 + 41 + 44 + 38 + 40 + 36 + 43}{10} = \frac{405}{10} = 40.5 \text{ thousand euros} $$

So, the point estimate of the average annual income in the population is 40.5 thousand euros.

To evaluate the reliability of the point estimate $ \bar{x} = 40.5 $, I can calculate the standard error using the sample standard deviation.

Note. We use the sample standard deviation because we don’t know the population standard deviation. In most cases, when working with sample data, we don’t have precise information about the entire population. If we did, we wouldn’t need to conduct a sample survey.

First, I calculate the sample standard deviation $ s $. Here’s how:

Find the deviations of the incomes from the sample mean (40.5):

$$ (35 - 40.5) = -5.5,\quad (42 - 40.5) = 1.5,\quad (39 - 40.5) = -1.5,\quad (47 - 40.5) = 6.5,\quad \text{etc.} $$

Square each deviation:

$$ (-5.5)^2 = 30.25,\quad (1.5)^2 = 2.25,\quad (-1.5)^2 = 2.25,\quad (6.5)^2 = 42.25,\quad \text{etc.} $$

Sum the squared deviations:

$$ 30.25 + 2.25 + 2.25 + 42.25 + 0.25 + 12.25 + 6.25 + 0.25 + 20.25 + 6.25 = 122.5 $$

Divide the sum by $ n - 1 $, where $ n $ is the sample size (in this case $ n = 10 $):

$$ s^2 = \frac{122.5}{10 - 1} = \frac{122.5}{9} \approx 13.61. $$

The sample standard deviation $ s $ is the square root of this value:

$$ s \approx \sqrt{13.61} \approx 3.69. $$

Now, I can calculate the standard error.

The standard error of the mean $ s_{\bar{x}} $ is calculated as:

$$ s_{\bar{x}} = \frac{s}{\sqrt{n}} = \frac{3.69}{\sqrt{10}} \approx \frac{3.69}{3.16} \approx 1.17 $$

Once the standard error is known, I can construct a confidence interval to estimate the population mean.

For example, let’s build a 95% confidence interval, which corresponds to a critical value $ z $ of approximately 1.96 for a normal distribution. The confidence interval is:

$$ \bar{x} \pm z \cdot s_{\bar{x}} = 40.5 \pm 1.96 \cdot 1.17 \approx 40.5 \pm 2.2932. $$

Thus, the 95% confidence interval for the population mean is approximately:

$$ [38.2068 \ , \ 42.7932] $$

This means that, with 95% confidence, the true population mean lies between 38.2 thousand euros and 42.79 thousand euros.

Note. In this example, I used a small sample of $ n=10 $ for simplicity, and applied a critical value $ z=1.96 $ to construct the 95% confidence interval. In reality, for sample sizes $ n < 30 $, the Central Limit Theorem doesn’t hold. Instead of using the critical values from a normal distribution, we should use the Student's t-distribution when working with small samples and the sample standard deviation $ s $. I simplified this for the sake of explanation.

Is one sample enough to estimate the population mean?

In general, a single sample can provide a good estimate of the population mean, but its reliability depends on several factors.

Here are a few key considerations:

Sample size
A small sample may not fully represent the population, and the mean estimate may be inaccurate. The larger the sample, the more likely the estimate is to be close to the true population mean. In statistics, the law of large numbers tells us that as the number of observations increases, the sample mean converges toward the population mean.
For instance, a sample of 10 observations may provide a useful estimate but might not capture all the characteristics of the population. A sample of 100 or 1,000 observations would offer much more accuracy.
Population variability
If there’s a lot of variability in the population (i.e., the values within the population differ greatly), it’s harder to get an accurate estimate of the mean with just one sample. In such cases, the standard error will be larger, and the confidence interval wider, indicating more uncertainty in the estimate. On the other hand, if the population values are more homogeneous, a moderate-sized sample might be sufficient to estimate the mean with reasonable accuracy.
Random sampling
It’s crucial that the sample is selected randomly. If the sample isn’t representative (for example, if there’s bias in how the observations were collected), the mean estimate could be skewed. Random sampling ensures that every individual in the population has an equal chance of being included in the sample, improving the quality of the estimate.
Confidence interval
While a single sample can provide a mean estimate, the confidence interval helps quantify the uncertainty around that estimate. If the confidence interval is too wide, it means the sample gives a less precise estimate, and it might be worth gathering a larger sample or more samples to obtain a more reliable estimate.
Repeated sampling
To improve the accuracy of the estimate, it’s helpful to take multiple independent samples from the same population. By calculating the sample means for each and then computing the mean of the sample means, we can get a more robust estimate of the population mean. This technique is known as bootstrapping or resampling.
In general, the distribution of sample means tends to approximate a normal distribution, regardless of the population distribution. Thus, working with the distribution of sample means is preferable because it allows us to make inferences based on the normality of the distribution, thanks to the Central Limit Theorem. It also reduces the impact of outliers and makes the population mean estimate more robust than relying on a single sample.

Overall, to obtain a more reliable estimate, it’s often preferable to work with larger samples or take multiple independent samples.

Additional notes

Some extra insights and considerations about point estimation of the mean:

Does the reliability of a single sample depend on the population distribution being normal?
The reliability of a single sample doesn’t necessarily require the population distribution to be normal, but the shape of the distribution can affect the accuracy and interpretation of results, especially when dealing with small samples.
- If the sample is large ( $ n \ge 30 $), the population distribution doesn’t need to be normal for us to obtain a reliable estimate of the mean, thanks to the Central Limit Theorem.
  Note. According to the Central Limit Theorem, no matter what the population distribution looks like, if the sample is large enough (usually $ n \ge 30 $), the distribution of the sample means will approach a normal distribution. Therefore, for large samples, the population doesn’t need to be normally distributed because the sample mean will be approximately normal.
- If the sample is small ( $ n < 30 $), the population distribution has a greater influence on the reliability of the mean estimate.
  - If the population distribution is highly non-normal, skewed, or has long tails, the sample mean might not be a good estimate of the population mean, and the sample mean distribution could deviate significantly from normality. In such cases, alternative methods or larger samples would be more appropriate.
  - If the population distribution is normal, even small samples can yield reliable mean estimates.

And so on.