Newborn Heights: Stats Analysis & Calculations

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Newborn Heights: Stats Analysis & Calculations

Hey guys! Let's dive into some stats related to the heights of newborn babies. We've got data from 20 little ones, and we're going to calculate some key statistical measures. So, buckle up, and let's get started!

Understanding Newborn Height Data

Before we jump into calculations, let's take a look at the dataset we're working with. We have the following heights (in cm) for 20 newborns:

41, 50, 52, 49, 54, 50, 47, 52, 49, 50, 52, 50, 47, 49, 51, 46, 50, 49, 50

Our goal is to find the mean, median, mode, variance, standard deviation, and coefficient of variation for this dataset. These measures will help us understand the central tendency, spread, and variability of newborn heights in our sample. Also, we assume that the population is normally distributed.

Why These Stats Matter

Understanding these statistical measures can provide valuable insights into the health and growth patterns of newborns. For example, the mean height gives us an average value, while the standard deviation tells us how much individual heights vary from this average. This information can be useful for healthcare professionals in assessing the overall health of a population of newborns and identifying any potential issues.

Calculating the Mean

The mean, also known as the average, is calculated by summing up all the values in the dataset and dividing by the number of values. In our case, we have 20 newborn heights. Here's the formula:

Mean = (Sum of all heights) / (Number of newborns)

Let's calculate the sum of the heights:

41 + 50 + 52 + 49 + 54 + 50 + 47 + 52 + 49 + 50 + 52 + 50 + 47 + 49 + 51 + 46 + 50 + 49 + 50 = 948

Now, we divide this sum by the number of newborns, which is 20:

Mean = 948 / 20 = 47.4 cm

So, the mean height of the newborns in our sample is 47.4 cm. This gives us a central point around which the data clusters.

Finding the Median

The median is the middle value in a dataset when the values are arranged in ascending order. If there's an even number of values (as in our case), the median is the average of the two middle values. First, let's sort the heights in ascending order:

41, 46, 47, 47, 49, 49, 49, 49, 50, 50, 50, 50, 50, 51, 52, 52, 52, 54

Since we have 20 values, the two middle values are the 10th and 11th values, which are both 50. To find the median, we take the average of these two values:

Median = (50 + 50) / 2 = 50 cm

Therefore, the median height of the newborns is 50 cm. The median is less sensitive to extreme values than the mean, making it a useful measure when the data may contain outliers.

Determining the Mode

The mode is the value that appears most frequently in a dataset. In our dataset of newborn heights, let's count the frequency of each height:

  • 41: 1
  • 46: 1
  • 47: 2
  • 49: 4
  • 50: 5
  • 51: 1
  • 52: 3
  • 54: 1

From the counts, we can see that the height 50 cm appears most frequently (5 times). Therefore, the mode of the newborn heights is 50 cm.

The mode gives us the most common height among the newborns in our sample. It's a useful measure for understanding the typical value in the dataset.

Calculating the Variance

The variance measures how spread out the data is from the mean. It's calculated by finding the average of the squared differences between each value and the mean. Here's the formula:

Variance = Σ(xi - mean)^2 / (n - 1)

Where:

  • xi is each individual height
  • mean is the mean height (47.4 cm)
  • n is the number of newborns (20)

First, we calculate the squared differences for each height:

  • (41 - 47.4)^2 = 40.96
  • (50 - 47.4)^2 = 6.76
  • (52 - 47.4)^2 = 21.16
  • (49 - 47.4)^2 = 2.56
  • (54 - 47.4)^2 = 43.56
  • (50 - 47.4)^2 = 6.76
  • (47 - 47.4)^2 = 0.16
  • (52 - 47.4)^2 = 21.16
  • (49 - 47.4)^2 = 2.56
  • (50 - 47.4)^2 = 6.76
  • (52 - 47.4)^2 = 21.16
  • (50 - 47.4)^2 = 6.76
  • (47 - 47.4)^2 = 0.16
  • (49 - 47.4)^2 = 2.56
  • (51 - 47.4)^2 = 12.96
  • (46 - 47.4)^2 = 1.96
  • (50 - 47.4)^2 = 6.76
  • (49 - 47.4)^2 = 2.56
  • (50 - 47.4)^2 = 6.76

Now, sum up all these squared differences:

  1. 96 + 6.76 + 21.16 + 2.56 + 43.56 + 6.76 + 0.16 + 21.16 + 2.56 + 6.76 + 21.16 + 6.76 + 0.16 + 2.56 + 12.96 + 1.96 + 6.76 + 2.56 + 6.76 = 233.2

Divide by (n - 1), which is (20 - 1) = 19:

Variance = 233.2 / 19 ≈ 12.27

So, the variance of the newborn heights is approximately 12.27 cm². The variance provides a measure of the overall spread of the data.

Calculating the Standard Deviation

The standard deviation is the square root of the variance. It gives us a measure of how much the individual values deviate from the mean in the original units (cm). Using the variance we calculated earlier (12.27 cm²), we can find the standard deviation:

Standard Deviation = √Variance

Standard Deviation = √12.27 ≈ 3.50 cm

Thus, the standard deviation of the newborn heights is approximately 3.50 cm. This tells us that, on average, the heights of the newborns in our sample deviate from the mean by about 3.50 cm.

Determining the Coefficient of Variation

The coefficient of variation (CV) is a relative measure of variability. It's calculated by dividing the standard deviation by the mean and multiplying by 100 to express it as a percentage. The formula is:

CV = (Standard Deviation / Mean) * 100

Using the standard deviation (3.50 cm) and the mean (47.4 cm) we calculated earlier, we can find the coefficient of variation:

CV = (3.50 / 47.4) * 100 ≈ 7.38%

Therefore, the coefficient of variation for the newborn heights is approximately 7.38%. This means that the standard deviation is about 7.38% of the mean. The CV is useful for comparing the variability of datasets with different units or different means.

Conclusion

Alright, folks! We've successfully calculated the mean, median, mode, variance, standard deviation, and coefficient of variation for the heights of 20 newborns. These measures give us a comprehensive understanding of the central tendency, spread, and variability of the data.

  • Mean: 47.4 cm
  • Median: 50 cm
  • Mode: 50 cm
  • Variance: 12.27 cm²
  • Standard Deviation: 3.50 cm
  • Coefficient of Variation: 7.38%

Understanding these statistical measures is super important for anyone working with data, whether it's in healthcare, finance, or any other field. Keep practicing, and you'll become a stats pro in no time!