Statistics Measures of Central Tendency

Definition: Central tendency measures the center or typical value of a dataset.
Common Measures:
- Mean: Average of all data points.
- Median: Middle value when data is sorted.
- Mode: Most frequently occurring value.

Definition: Quartiles divide a dataset into four equal parts.
Key Quartiles:
- Q1 (First Quartile): 25th percentile, median of the first half of data.
- Q2 (Second Quartile): 50th percentile, median of the dataset.
- Q3 (Third Quartile): 75th percentile, median of the second half of data.
Calculation Steps:
1. Sort the data in ascending order.
2. Find the median (Q2).
3. Split the data into two halves (lower and upper).
4. Calculate Q1 from the lower half and Q3 from the upper half.

Definition: Deciles divide a dataset into ten equal parts.
Key Deciles:
- D1: 10th percentile
- D2: 20th percentile
- D3: 30th percentile
- …
- D9: 90th percentile
Calculation Steps:
1. Sort data in ascending order.
2. Use the formula: ( D_k = \frac{k(n+1)}{10} ), where ( k ) is the decile number (1-9), and ( n ) is the number of data points.
3. If the result is a whole number, the decile is the average of the values at that position and the next. If not, round to the nearest whole number.

Definition: Describes the way values are spread or clustered in a dataset.
Common Types:
- Uniform Distribution: All values are equally likely.
- Normal Distribution: Bell-shaped curve; mean, median, and mode are equal.
- Skewed Distribution: Values are not symmetrically distributed (left or right skew).

Definition: Percentiles indicate the relative standing of a value in a dataset.
Key Percentiles:
- P1 (1st Percentile): 1% of values fall below this point.
- P50 (50th Percentile): Median of the dataset.
- P99 (99th Percentile): 99% of values fall below this point.
Calculation Steps:
1. Sort the data in ascending order.
2. Use the formula: ( P_k = \frac{k(n+1)}{100} ), where ( k ) is the desired percentile (0-100), and ( n ) is the number of data points.
3. Similar to deciles, if the result is a whole number, average the values at that position and the next; if not, round to the nearest whole number.

Measures of central tendency provide insight into the center or typical value of data.
The mean is calculated by summing all data points and dividing by the count of values.
The median is found by sorting data and selecting the middle value; applicable in both odd and even datasets.
The mode represents the value that appears most frequently within the dataset.

Quartiles segment a dataset into four equal parts, essential for understanding data spread.
Q1 (First Quartile) corresponds to the 25th percentile, marking the median in the first half of data.
Q2 (Second Quartile), also known as the median, represents the 50th percentile and divides the dataset into two equal halves.
Q3 (Third Quartile) indicates the 75th percentile, marking the median in the second half of the data.
To calculate quartiles: sort data, determine Q2, then split data to compute Q1 and Q3.

Deciles partition a dataset into ten equal parts, enhancing detailed data analysis.
Each decile is identified by its corresponding percentile: D1 (10%), D2 (20%), D3 (30%), up to D9 (90%).
The decile formula is expressed as ( D_k = \frac{k(n+1)}{10} ); where ( k ) is the decile's index and ( n ) is the total data points.
To find a decile, if the result is whole, average the values at that index; round if the result is not whole.

Data distribution illustrates how values are dispersed or grouped within a set.
Uniform Distribution suggests equal likelihood for all data points to occur.
Normal Distribution features a symmetrical bell shape, where mean, median, and mode coincide.
A Skewed Distribution indicates asymmetry; data may lean left or right, affecting the central tendency measures.

Percentiles reveal the relative position of a specific value within an ordered dataset.
P1 (1st Percentile) captures the lowest 1% of values, while P50 (50th Percentile) represents the median.
The P99 (99th Percentile) indicates that 99% of values fall below this threshold.
Percentile calculations involve sorting data and applying the formula ( P_k = \frac{k(n+1)}{100} ); similar rounding rules as deciles apply for determining precise percentile values.

Quartiles split a dataset into four equal parts, helping to quantify the spread of data.
Q1 (First Quartile) marks the 25th percentile and is the median of the lower half of data.
Q2 (Second Quartile) is the median of the dataset, representing the 50th percentile.
Q3 (Third Quartile) is the median of the upper half of data, falling at the 75th percentile.
To find quartiles:
- Sort data in ascending order.
- Identify the median (Q2).
- Divide data into lower and upper halves.
- Calculate Q1 from the lower half and Q3 from the upper half.

Deciles divide a dataset into ten equal parts, providing insights into data distribution.
D1 through D9 represent the 10th to 90th percentiles, respectively.
Calculation of deciles involves:
- Sorting the data in ascending order.
- Using the formula ( D_k = \frac{k(n+1)}{10} ), where ( k ) is the decile number and ( n ) is the number of data points.
- If the result is whole, the decile is the average of the values at that position and the next; otherwise, round to the nearest whole number.

Data distribution describes how values are spread or clustered.
Uniform distribution indicates all values are equally likely to occur.
Normal distribution forms a bell-shaped curve, characterized by equal mean, median, and mode.
Skewed distribution shows asymmetrical data spread, with left or right skewness.

Percentiles express the relative standing of a value within a dataset.
P1 (1st Percentile) signifies that 1% of values fall below this point.
P50 (50th Percentile) represents the median value of the dataset.
P99 (99th Percentile) indicates that 99% of values fall below this threshold.
Percentiles are calculated by:
- Sorting data in ascending order.
- Applying the formula ( P_k = \frac{k(n+1)}{100} ), where ( k ) is the desired percentile and ( n ) is the sample size.
- The same rounding rules for deciles apply, averaging values at that position if the result is whole.

Range denotes the difference between the maximum and minimum values.
Quartile deviation measures the spread of the middle half of the data.
Interquartile range (IQR) is the difference between Q3 and Q1, representing the range of the middle 50% of the data.
Standard deviation quantifies the variation or dispersion of data points around the mean.
Coefficient of variation (CV) compares the standard deviation to the mean, expressed as a percentage, allowing for comparison across different datasets.