Catriona: /* Notes: Key concepts for descriptive statistics */

2014-07-25T06:51:42Z

‎Notes: Key concepts for descriptive statistics

Catriona at 06:50, 25 July 2014

2014-07-25T06:50:39Z

Catriona: Created page with "An important thing to do is to describe the variable of interest. A variable is any characteristics, number, or quantity that can be measured or counted. Jam untuk menyiasat i..."

2014-07-25T06:49:47Z

Created page with "An important thing to do is to describe the variable of interest. A variable is any characteristics, number, or quantity that can be measured or counted. Jam untuk menyiasat i..."

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An important thing to do is to describe the variable of interest. A variable is any characteristics, number, or quantity that can be measured or counted. Jam untuk menyiasat is a continuous variable.

''' Measures of central tendency (mean and median)'''
The first step in describing the variable jam untuk menyiasat is to estimate the most common value. There are several ways of doing this, but the best ways are usually to measure the mean and the median.

'''''Mean'''''

The mean estimates the centre of the all the values for jam untuk menyiasat. You can estimate the mean by summing all your values of jam untuk menyiasat and dividing the sum by the number of values.

The formula for the mean is:

<math>\bar{x}=\frac{\sum x}{n}</math>

Where <math>\bar{x}</math>= the sample mean, ∑ = symbol for sum, χ = the variable we are interested in (here jam untuk menyiasat) and n = the number of observations.

For example:

We have 5 observations of the variable jam untuk menyiasat (24, 0, 48, 78 and 96 hours).

The mean is:
<math>\bar{x}=\frac{24+0+48+78+96}{5}= 49.2</math>

We would expect it to take 49.2 hours to investigate a report of a priority syndrome.

'''''Median '''''

The median measures the middle score. The median is less affected by outliers (extreme values that are not very similar to other observations of a variable) than the mean. To calculate the median we order all the measurements from lowest to highest and take the middle score as the median. Hence, the ordered list is:

0, 24, 48, 78, 96 hours.

The middle score is 48 hours. In this case the median is similar to the mean. Often the mean and median are similar if the data is normally distributed or the sample is large.

Later we will calculate the mean and median from the real data using Excel.

''' Measures of spread (range, interquartile range, standard deviation and confidence intervals)'''

It is important to understand how much the measurements of the variable are spread around the mean or median that you have calculated. A measure of spread helps you to understand how good your sample is at estimating the mean or median. If the sample is large or the variability in observations is low, the measure of spread will be smaller. This indicates you can have higher confidence that the estimated mean or median is close to the real value.

Generally, we use standard deviations or confidence intervals with means. For medians it is common to calculate an interquartile range. For both mean and median, a range is useful.

'''''Range'''''

The range is the difference between the highest and lowest observation: <math> 96-0=96 \mathrm{~hours.}</math>

'''''Interquartile range'''''

The interquartile range is the difference between the upper and lower quartiles. The upper quartile (Q3) is the middle measurement between the median and highest value. The lower quartile (Q1) is the middle value between the median and the lowest value. 25% of values occur in the each of the upper and lower quartile and 50% of values occur within the interquartile range.

For example for the ordered 5 simulated measurements of the jam untuk menyiasat:}}

{| class="wikitable"|center
|-
! '''Measurement (hours) !! Parameter
|-
| 0 || Lowest measurement
|-
| 24 || Q1 (lower quartile)
|-
| 48 || Median (middle measurement)
|-
| 78 || Q3 (upper quartile)
|-
| 96 || Highest measurement

|}

{{hlbox|The interquartile range is:

<math> \mathrm{Q3-Q1~}=\mathrm{~78-24~=~54~hours}</math>

This is interpreted as the middle 50% of the measurements are spread across 54 hours from 24 to 78 hours jam untuk menyiasat.

'''''Standard deviation'''''

The standard deviation (SD) measures the average of the differences of each observation from the mean. We calculate the SD for jam untuk menyiasat as:

SD = <math>\sqrt{\frac{\sum {\left({x} - \bar{x}\right)}^{2}}{n}}=\sqrt{\frac{{\left(0-49.2\right)}^{2}+{\left(24-49.2\right)}^{2}+{\left(48-49.2\right)}^{2}+{\left(78-49.2\right)}^{2}+{\left(96-49.2\right)}^{2}}{5}}=\sqrt{\frac{6077}{5}}=\sqrt{1215}=35</math>

Where <math>\mathrm{x = 0, 24, 48, 78~and~96~hours,~n = 5,~\bar{x}=49.2~hours~and~ SD = standard~deviation}.</math>

Here we interpret this as: on average each observation of jam untuk menyiasat will differ from the mean of 49.2 hours by 35 hours. There is a lot of variability in the data. We are not very sure that the mean of 49.2 hours will be close to each observation.

'''''Confidence interval'''''

The confidence interval is an interval of plausible values around an estimate. A 95% confidence interval can be calculated as follows:

<math>\mathrm{95%~CI}=\bar{x}\pm Z \mathrm{~SD}</math>

where Z = 1.96, SD =standard deviation and <math>\bar{x}</math> is the estimated mean.

<math>\mathrm{95%~CI}=\bar{x}\pm 1.96\mathrm{SD}=49.2\pm 1.96\left(35\right)=49.2\pm 69=-20 \mathrm{~to~}118 \mathrm{~hours}</math>

Thus our confidence interval for jam untuk menyiasat is from -20 to 118 hours. -20 is impossible (we cannot investigate before a report!) so we would accept 0 hours instead.

We interpret this as follows.

We are 95% confident that the true population mean of jam untuk menyiasat lies between -20 (0) and 118 hours. So our sampling is not good enough to estimate the mean very accurately because the confidence interval is quite wide.

More formally, if we repeated our sampling from the population in the same manner many times the true population mean will be between -20 (0) hours and 118 hours 95% of the time.

Confidence intervals can be calculated for many other measures.

← Revisi sebelumnya		Revisi per 25 Juli 2014 06.50
Baris 1:		Baris 1:
		+	===Notes: Key concepts for descriptive statistics===
		+
	An important thing to do is to describe the variable of interest. A variable is any characteristics, number, or quantity that can be measured or counted. Jam untuk menyiasat is a continuous variable.		An important thing to do is to describe the variable of interest. A variable is any characteristics, number, or quantity that can be measured or counted. Jam untuk menyiasat is a continuous variable.

@@ Baris 49: / Baris 49: @@
 The interquartile range is the difference between the upper and lower quartiles. The upper quartile (Q3) is the middle measurement between the median and highest value. The lower quartile (Q1) is the middle value between the median and the lowest value. 25% of values occur in the each of the upper and lower quartile and 50% of values occur within the interquartile range.
-For example for the ordered 5 simulated measurements of the jam untuk menyiasat:}}
+For example for the ordered 5 simulated measurements of the jam untuk menyiasat:
-{| class="wikitable"|center
+{| class="wikitable"|
 |-
 ! '''Measurement (hours) !! Parameter
@@ Baris 67: / Baris 67: @@
 |}
-{{hlbox|The interquartile range is:
+The interquartile range is:
 <math> \mathrm{Q3-Q1~}=\mathrm{~78-24~=~54~hours}</math>

Data Analysis:Notes Descriptive statistics - Revision history

Catriona: /* Notes: Key concepts for descriptive statistics */

Catriona at 06:50, 25 July 2014

Catriona: Created page with "An important thing to do is to describe the variable of interest. A variable is any characteristics, number, or quantity that can be measured or counted. Jam untuk menyiasat i..."